Calculus Examples

$y = \frac{(15 x - x^{8})}{5}$ , $[- 5, 5]$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$\frac{15 x - x^{8}}{5} = 0$

Step 1.2

Solve $\frac{15 x - x^{8}}{5} = 0$ for $x$ .

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Step 1.2.1

Set the numerator equal to zero.

$15 x - x^{8} = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Factor $x$ out of $15 x - x^{8}$ .

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Step 1.2.2.1.1

Factor $x$ out of $15 x$ .

$x \cdot 15 - x^{8} = 0$

Step 1.2.2.1.2

Factor $x$ out of $- x^{8}$ .

$x \cdot 15 + x (- x^{7}) = 0$

Step 1.2.2.1.3

Factor $x$ out of $x \cdot 15 + x (- x^{7})$ .

$x (15 - x^{7}) = 0$

Step 1.2.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$15 - x^{7} = 0 + y = 0$

Step 1.2.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.2.4

Set $15 - x^{7}$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $15 - x^{7}$ equal to $0$ .

$15 - x^{7} = 0$

Step 1.2.2.4.2

Solve $15 - x^{7} = 0$ for $x$ .

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Step 1.2.2.4.2.1

Subtract $15$ from both sides of the equation.

$- x^{7} = - 15$

Step 1.2.2.4.2.2

Divide each term in $- x^{7} = - 15$ by $- 1$ and simplify.

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Step 1.2.2.4.2.2.1

Divide each term in $- x^{7} = - 15$ by $- 1$ .

$\frac{- x^{7}}{- 1} = \frac{- 15}{- 1}$

Step 1.2.2.4.2.2.2

Simplify the left side.

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Step 1.2.2.4.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x^{7}}{1} = \frac{- 15}{- 1}$

Step 1.2.2.4.2.2.2.2

Divide $x^{7}$ by $1$ .

$x^{7} = \frac{- 15}{- 1}$

Step 1.2.2.4.2.2.3

Simplify the right side.

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Step 1.2.2.4.2.2.3.1

Divide $- 15$ by $- 1$ .

$x^{7} = 15$

Step 1.2.2.4.2.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[7]{15}$

Step 1.2.2.5

The final solution is all the values that make $x (15 - x^{7}) = 0$ true.

$x = 0, \sqrt[7]{15}$

Step 1.3

Substitute $0$ for $x$ .

$y = 0$

Step 1.4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 0)$

$(\sqrt[7]{15}, 0)$

$(0, 0)$

$(\sqrt[7]{15}, 0)$

Step 2

Factor $x$ out of $15 x - x^{8}$ .

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Step 2.1

Factor $x$ out of $15 x$ .

$y = \frac{x \cdot 15 - x^{8}}{5}$

Step 2.2

Factor $x$ out of $- x^{8}$ .

$y = \frac{x \cdot 15 + x (- x^{7})}{5}$

Step 2.3

Factor $x$ out of $x \cdot 15 + x (- x^{7})$ .

$y = \frac{x (15 - x^{7})}{5}$

Step 3

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{- 5}^{0} 0 d x - \int_{- 5}^{0} \frac{x (15 - x^{7})}{5} d x$

Step 4

Integrate to find the area between $- 5$ and $0$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{- 5}^{0} 0 - (\frac{x (15 - x^{7})}{5}) d x$

Step 4.2

Subtract $\frac{x (15 - x^{7})}{5}$ from $0$ .

$\int_{- 5}^{0} - \frac{x (15 - x^{7})}{5} d x$

Step 4.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int_{- 5}^{0} \frac{x (15 - x^{7})}{5} d x$

Step 4.4

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$- (\frac{1}{5} \int_{- 5}^{0} x (15 - x^{7}) d x)$

Step 4.5

Multiply $x (15 - x^{7})$ .

$- \frac{1}{5} \int_{- 5}^{0} x \cdot 15 + x (- x^{7}) d x$

Step 4.6

Simplify.

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Step 4.6.1

Move $15$ to the left of $x$ .

$- \frac{1}{5} \int_{- 5}^{0} 15 \cdot x + x (- x^{7}) d x$

Step 4.6.2

Raise $x$ to the power of $1$ .

$- \frac{1}{5} \int_{- 5}^{0} 15 x - (x^{1} x^{7}) d x$

Step 4.6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{5} \int_{- 5}^{0} 15 x - x^{1 + 7} d x$

Step 4.6.4

Add $1$ and $7$ .

$- \frac{1}{5} \int_{- 5}^{0} 15 x - x^{8} d x$

Step 4.7

Split the single integral into multiple integrals.

$- \frac{1}{5} (\int_{- 5}^{0} 15 x d x + \int_{- 5}^{0} - x^{8} d x)$

Step 4.8

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$- \frac{1}{5} (15 \int_{- 5}^{0} x d x + \int_{- 5}^{0} - x^{8} d x)$

Step 4.9

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- \frac{1}{5} (15 (\frac{1}{2} x^{2}]_{- 5}^{0}) + \int_{- 5}^{0} - x^{8} d x)$

Step 4.10

Combine $\frac{1}{2}$ and $x^{2}$ .

$- \frac{1}{5} (15 (\frac{x^{2}}{2}]_{- 5}^{0}) + \int_{- 5}^{0} - x^{8} d x)$

Step 4.11

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \frac{1}{5} (15 (\frac{x^{2}}{2}]_{- 5}^{0}) - \int_{- 5}^{0} x^{8} d x)$

Step 4.12

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$- \frac{1}{5} (15 (\frac{x^{2}}{2}]_{- 5}^{0}) - (\frac{1}{9} x^{9}]_{- 5}^{0}))$

Step 4.13

Simplify the answer.

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Step 4.13.1

Combine $\frac{1}{9}$ and $x^{9}$ .

$- \frac{1}{5} (15 (\frac{x^{2}}{2}]_{- 5}^{0}) - (\frac{x^{9}}{9}]_{- 5}^{0}))$

Step 4.13.2

Substitute and simplify.

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Step 4.13.2.1

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 5$ .

$- \frac{1}{5} (15 ((\frac{0^{2}}{2}) - \frac{{(- 5)}^{2}}{2}) - (\frac{x^{9}}{9}]_{- 5}^{0}))$

Step 4.13.2.2

Evaluate $\frac{x^{9}}{9}$ at $0$ and at $- 5$ .

$- \frac{1}{5} (15 (\frac{0^{2}}{2} - \frac{{(- 5)}^{2}}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3

Simplify.

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Step 4.13.2.3.1

Raising $0$ to any positive power yields $0$ .

$- \frac{1}{5} (15 (\frac{0}{2} - \frac{{(- 5)}^{2}}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.2

Cancel the common factor of $0$ and $2$ .

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Step 4.13.2.3.2.1

Factor $2$ out of $0$ .

$- \frac{1}{5} (15 (\frac{2 (0)}{2} - \frac{{(- 5)}^{2}}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.2.2

Cancel the common factors.

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Step 4.13.2.3.2.2.1

Factor $2$ out of $2$ .

$- \frac{1}{5} (15 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 5)}^{2}}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.2.2.2

Cancel the common factor.

$- \frac{1}{5} (15 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 5)}^{2}}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.2.2.3

Rewrite the expression.

$- \frac{1}{5} (15 (\frac{0}{1} - \frac{{(- 5)}^{2}}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.2.2.4

Divide $0$ by $1$ .

$- \frac{1}{5} (15 (0 - \frac{{(- 5)}^{2}}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.3

Raise $- 5$ to the power of $2$ .

$- \frac{1}{5} (15 (0 - \frac{25}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.4

Subtract $\frac{25}{2}$ from $0$ .

$- \frac{1}{5} (15 (- \frac{25}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.5

Multiply $- 1$ by $15$ .

$- \frac{1}{5} (- 15 (\frac{25}{2}) - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.6

Combine $- 15$ and $\frac{25}{2}$ .

$- \frac{1}{5} (\frac{- 15 \cdot 25}{2} - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.7

Multiply $- 15$ by $25$ .

$- \frac{1}{5} (\frac{- 375}{2} - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.8

Move the negative in front of the fraction.

$- \frac{1}{5} (- \frac{375}{2} - (\frac{0^{9}}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.9

Raising $0$ to any positive power yields $0$ .

$- \frac{1}{5} (- \frac{375}{2} - (\frac{0}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.10

Cancel the common factor of $0$ and $9$ .

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Step 4.13.2.3.10.1

Factor $9$ out of $0$ .

$- \frac{1}{5} (- \frac{375}{2} - (\frac{9 (0)}{9} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.10.2

Cancel the common factors.

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Step 4.13.2.3.10.2.1

Factor $9$ out of $9$ .

$- \frac{1}{5} (- \frac{375}{2} - (\frac{9 \cdot 0}{9 \cdot 1} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.10.2.2

Cancel the common factor.

$- \frac{1}{5} (- \frac{375}{2} - (\frac{9 \cdot 0}{9 \cdot 1} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.10.2.3

Rewrite the expression.

$- \frac{1}{5} (- \frac{375}{2} - (\frac{0}{1} - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.10.2.4

Divide $0$ by $1$ .

$- \frac{1}{5} (- \frac{375}{2} - (0 - \frac{{(- 5)}^{9}}{9}))$

Step 4.13.2.3.11

Raise $- 5$ to the power of $9$ .

$- \frac{1}{5} (- \frac{375}{2} - (0 - \frac{- 1953125}{9}))$

Step 4.13.2.3.12

Move the negative in front of the fraction.

$- \frac{1}{5} (- \frac{375}{2} - (0 - - \frac{1953125}{9}))$

Step 4.13.2.3.13

Multiply $- 1$ by $- 1$ .

$- \frac{1}{5} (- \frac{375}{2} - (0 + 1 (\frac{1953125}{9})))$

Step 4.13.2.3.14

Multiply $\frac{1953125}{9}$ by $1$ .

$- \frac{1}{5} (- \frac{375}{2} - (0 + \frac{1953125}{9}))$

Step 4.13.2.3.15

Add $0$ and $\frac{1953125}{9}$ .

$- \frac{1}{5} (- \frac{375}{2} - \frac{1953125}{9})$

Step 4.13.2.3.16

To write $- \frac{375}{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{1}{5} (- \frac{375}{2} \cdot \frac{9}{9} - \frac{1953125}{9})$

Step 4.13.2.3.17

To write $- \frac{1953125}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{5} (- \frac{375}{2} \cdot \frac{9}{9} - \frac{1953125}{9} \cdot \frac{2}{2})$

Step 4.13.2.3.18

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.13.2.3.18.1

Multiply $\frac{375}{2}$ by $\frac{9}{9}$ .

$- \frac{1}{5} (- \frac{375 \cdot 9}{2 \cdot 9} - \frac{1953125}{9} \cdot \frac{2}{2})$

Step 4.13.2.3.18.2

Multiply $2$ by $9$ .

$- \frac{1}{5} (- \frac{375 \cdot 9}{18} - \frac{1953125}{9} \cdot \frac{2}{2})$

Step 4.13.2.3.18.3

Multiply $\frac{1953125}{9}$ by $\frac{2}{2}$ .

$- \frac{1}{5} (- \frac{375 \cdot 9}{18} - \frac{1953125 \cdot 2}{9 \cdot 2})$

Step 4.13.2.3.18.4

Multiply $9$ by $2$ .

$- \frac{1}{5} (- \frac{375 \cdot 9}{18} - \frac{1953125 \cdot 2}{18})$

Step 4.13.2.3.19

Combine the numerators over the common denominator.

$- \frac{1}{5} \cdot \frac{- 375 \cdot 9 - 1953125 \cdot 2}{18}$

Step 4.13.2.3.20

Simplify the numerator.

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Step 4.13.2.3.20.1

Multiply $- 375$ by $9$ .

$- \frac{1}{5} \cdot \frac{- 3375 - 1953125 \cdot 2}{18}$

Step 4.13.2.3.20.2

Multiply $- 1953125$ by $2$ .

$- \frac{1}{5} \cdot \frac{- 3375 - 3906250}{18}$

Step 4.13.2.3.20.3

Subtract $3906250$ from $- 3375$ .

$- \frac{1}{5} \cdot \frac{- 3909625}{18}$

Step 4.13.2.3.21

Move the negative in front of the fraction.

$- \frac{1}{5} (- \frac{3909625}{18})$

Step 4.13.2.3.22

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{5}) \frac{3909625}{18}$

Step 4.13.2.3.23

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{5} \cdot \frac{3909625}{18}$

Step 4.13.2.3.24

Multiply $\frac{1}{5}$ by $\frac{3909625}{18}$ .

$\frac{3909625}{5 \cdot 18}$

Step 4.13.2.3.25

Multiply $5$ by $18$ .

$\frac{3909625}{90}$

Step 4.13.2.3.26

Cancel the common factor of $3909625$ and $90$ .

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Step 4.13.2.3.26.1

Factor $5$ out of $3909625$ .

$\frac{5 (781925)}{90}$

Step 4.13.2.3.26.2

Cancel the common factors.

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Step 4.13.2.3.26.2.1

Factor $5$ out of $90$ .

$\frac{5 \cdot 781925}{5 \cdot 18}$

Step 4.13.2.3.26.2.2

Cancel the common factor.

$\frac{5 \cdot 781925}{5 \cdot 18}$

Step 4.13.2.3.26.2.3

Rewrite the expression.

$\frac{781925}{18}$

Step 5

$A r e a = \int_{0}^{\sqrt[7]{15}} \frac{x (15 - x^{7})}{5} d x - \int_{0}^{\sqrt[7]{15}} 0 d x$

Step 6

Integrate to find the area between $0$ and $\sqrt[7]{15}$ .

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Step 6.1

Combine the integrals into a single integral.

$\int_{0}^{\sqrt[7]{15}} \frac{x (15 - x^{7})}{5} - (0) d x$

Step 6.2

Subtract $0$ from $\frac{x (15 - x^{7})}{5}$ .

$\int_{0}^{\sqrt[7]{15}} \frac{x (15 - x^{7})}{5} d x$

Step 6.3

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int_{0}^{\sqrt[7]{15}} x (15 - x^{7}) d x$

Step 6.4

Multiply $x (15 - x^{7})$ .

$\frac{1}{5} \int_{0}^{\sqrt[7]{15}} x \cdot 15 + x (- x^{7}) d x$

Step 6.5

Simplify.

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Step 6.5.1

Move $15$ to the left of $x$ .

$\frac{1}{5} \int_{0}^{\sqrt[7]{15}} 15 \cdot x + x (- x^{7}) d x$

Step 6.5.2

Raise $x$ to the power of $1$ .

$\frac{1}{5} \int_{0}^{\sqrt[7]{15}} 15 x - (x^{1} x^{7}) d x$

Step 6.5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{5} \int_{0}^{\sqrt[7]{15}} 15 x - x^{1 + 7} d x$

Step 6.5.4

Add $1$ and $7$ .

$\frac{1}{5} \int_{0}^{\sqrt[7]{15}} 15 x - x^{8} d x$

Step 6.6

Split the single integral into multiple integrals.

$\frac{1}{5} (\int_{0}^{\sqrt[7]{15}} 15 x d x + \int_{0}^{\sqrt[7]{15}} - x^{8} d x)$

Step 6.7

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$\frac{1}{5} (15 \int_{0}^{\sqrt[7]{15}} x d x + \int_{0}^{\sqrt[7]{15}} - x^{8} d x)$

Step 6.8

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$\frac{1}{5} (15 (\frac{1}{2} x^{2}]_{0}^{\sqrt[7]{15}}) + \int_{0}^{\sqrt[7]{15}} - x^{8} d x)$

Step 6.9

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{1}{5} (15 (\frac{x^{2}}{2}]_{0}^{\sqrt[7]{15}}) + \int_{0}^{\sqrt[7]{15}} - x^{8} d x)$

Step 6.10

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{5} (15 (\frac{x^{2}}{2}]_{0}^{\sqrt[7]{15}}) - \int_{0}^{\sqrt[7]{15}} x^{8} d x)$

Step 6.11

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$\frac{1}{5} (15 (\frac{x^{2}}{2}]_{0}^{\sqrt[7]{15}}) - (\frac{1}{9} x^{9}]_{0}^{\sqrt[7]{15}}))$

Step 6.12

Simplify the answer.

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Step 6.12.1

Combine $\frac{1}{9}$ and $x^{9}$ .

$\frac{1}{5} (15 (\frac{x^{2}}{2}]_{0}^{\sqrt[7]{15}}) - (\frac{x^{9}}{9}]_{0}^{\sqrt[7]{15}}))$

Step 6.12.2

Substitute and simplify.

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Step 6.12.2.1

Evaluate $\frac{x^{2}}{2}$ at $\sqrt[7]{15}$ and at $0$ .

$\frac{1}{5} (15 ((\frac{{\sqrt[7]{15}}^{2}}{2}) - \frac{0^{2}}{2}) - (\frac{x^{9}}{9}]_{0}^{\sqrt[7]{15}}))$

Step 6.12.2.2

Evaluate $\frac{x^{9}}{9}$ at $\sqrt[7]{15}$ and at $0$ .

$\frac{1}{5} (15 (\frac{{\sqrt[7]{15}}^{2}}{2} - \frac{0^{2}}{2}) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3

Simplify.

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Step 6.12.2.3.1

Rewrite ${\sqrt[7]{15}}^{2}$ as $\sqrt[7]{15^{2}}$ .

$\frac{1}{5} (15 (\frac{\sqrt[7]{15^{2}}}{2} - \frac{0^{2}}{2}) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.2

Raise $15$ to the power of $2$ .

$\frac{1}{5} (15 (\frac{\sqrt[7]{225}}{2} - \frac{0^{2}}{2}) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.3

Raising $0$ to any positive power yields $0$ .

$\frac{1}{5} (15 (\frac{\sqrt[7]{225}}{2} - \frac{0}{2}) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.4

Cancel the common factor of $0$ and $2$ .

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Step 6.12.2.3.4.1

Factor $2$ out of $0$ .

$\frac{1}{5} (15 (\frac{\sqrt[7]{225}}{2} - \frac{2 (0)}{2}) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.4.2

Cancel the common factors.

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Step 6.12.2.3.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{5} (15 (\frac{\sqrt[7]{225}}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.4.2.2

Cancel the common factor.

$\frac{1}{5} (15 (\frac{\sqrt[7]{225}}{2} - \frac{2 \cdot 0}{2 \cdot 1}) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.4.2.3

Rewrite the expression.

$\frac{1}{5} (15 (\frac{\sqrt[7]{225}}{2} - \frac{0}{1}) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.4.2.4

Divide $0$ by $1$ .

$\frac{1}{5} (15 (\frac{\sqrt[7]{225}}{2} - 0) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.5

Multiply $- 1$ by $0$ .

$\frac{1}{5} (15 (\frac{\sqrt[7]{225}}{2} + 0) - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.6

Add $\frac{\sqrt[7]{225}}{2}$ and $0$ .

$\frac{1}{5} (15 \frac{\sqrt[7]{225}}{2} - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.7

Combine $15$ and $\frac{\sqrt[7]{225}}{2}$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{{\sqrt[7]{15}}^{9}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.8

Rewrite ${\sqrt[7]{15}}^{9}$ as $\sqrt[7]{15^{9}}$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{15^{9}}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.9

Raise $15$ to the power of $9$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{38443359375}}{9} - \frac{0^{9}}{9}))$

Step 6.12.2.3.10

Raising $0$ to any positive power yields $0$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{38443359375}}{9} - \frac{0}{9}))$

Step 6.12.2.3.11

Cancel the common factor of $0$ and $9$ .

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Step 6.12.2.3.11.1

Factor $9$ out of $0$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{38443359375}}{9} - \frac{9 (0)}{9}))$

Step 6.12.2.3.11.2

Cancel the common factors.

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Step 6.12.2.3.11.2.1

Factor $9$ out of $9$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{38443359375}}{9} - \frac{9 \cdot 0}{9 \cdot 1}))$

Step 6.12.2.3.11.2.2

Cancel the common factor.

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{38443359375}}{9} - \frac{9 \cdot 0}{9 \cdot 1}))$

Step 6.12.2.3.11.2.3

Rewrite the expression.

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{38443359375}}{9} - \frac{0}{1}))$

Step 6.12.2.3.11.2.4

Divide $0$ by $1$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{38443359375}}{9} - 0))$

Step 6.12.2.3.12

Multiply $- 1$ by $0$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - (\frac{\sqrt[7]{38443359375}}{9} + 0))$

Step 6.12.2.3.13

Add $\frac{\sqrt[7]{38443359375}}{9}$ and $0$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - \frac{\sqrt[7]{38443359375}}{9})$

Step 6.12.3

Simplify.

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Step 6.12.3.1

Rewrite $38443359375$ as $15^{7} \cdot 225$ .

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Step 6.12.3.1.1

Factor $170859375$ out of $38443359375$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - \frac{\sqrt[7]{170859375 (225)}}{9})$

Step 6.12.3.1.2

Rewrite $170859375$ as $15^{7}$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - \frac{\sqrt[7]{15^{7} \cdot 225}}{9})$

Step 6.12.3.2

Pull terms out from under the radical.

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - \frac{15 \sqrt[7]{225}}{9})$

Step 6.12.3.3

Cancel the common factor of $15$ and $9$ .

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Step 6.12.3.3.1

Factor $3$ out of $15 \sqrt[7]{225}$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - \frac{3 (5 \sqrt[7]{225})}{9})$

Step 6.12.3.3.2

Cancel the common factors.

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Step 6.12.3.3.2.1

Factor $3$ out of $9$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - \frac{3 (5 \sqrt[7]{225})}{3 (3)})$

Step 6.12.3.3.2.2

Cancel the common factor.

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - \frac{3 (5 \sqrt[7]{225})}{3 \cdot 3})$

Step 6.12.3.3.2.3

Rewrite the expression.

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} - \frac{5 \sqrt[7]{225}}{3})$

Step 6.12.3.4

To write $\frac{15 \sqrt[7]{225}}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} \cdot \frac{3}{3} - \frac{5 \sqrt[7]{225}}{3})$

Step 6.12.3.5

To write $- \frac{5 \sqrt[7]{225}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225}}{2} \cdot \frac{3}{3} - \frac{5 \sqrt[7]{225}}{3} \cdot \frac{2}{2})$

Step 6.12.3.6

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.12.3.6.1

Multiply $\frac{15 \sqrt[7]{225}}{2}$ by $\frac{3}{3}$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225} \cdot 3}{2 \cdot 3} - \frac{5 \sqrt[7]{225}}{3} \cdot \frac{2}{2})$

Step 6.12.3.6.2

Multiply $2$ by $3$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225} \cdot 3}{6} - \frac{5 \sqrt[7]{225}}{3} \cdot \frac{2}{2})$

Step 6.12.3.6.3

Multiply $\frac{5 \sqrt[7]{225}}{3}$ by $\frac{2}{2}$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225} \cdot 3}{6} - \frac{5 \sqrt[7]{225} \cdot 2}{3 \cdot 2})$

Step 6.12.3.6.4

Multiply $3$ by $2$ .

$\frac{1}{5} (\frac{15 \sqrt[7]{225} \cdot 3}{6} - \frac{5 \sqrt[7]{225} \cdot 2}{6})$

Step 6.12.3.7

Combine the numerators over the common denominator.

$\frac{1}{5} \cdot \frac{15 \sqrt[7]{225} \cdot 3 - 5 \sqrt[7]{225} \cdot 2}{6}$

Step 6.12.3.8

Multiply $3$ by $15$ .

$\frac{1}{5} \cdot \frac{45 \sqrt[7]{225} - 5 \sqrt[7]{225} \cdot 2}{6}$

Step 6.12.3.9

Multiply $2$ by $- 5$ .

$\frac{1}{5} \cdot \frac{45 \sqrt[7]{225} - 10 \sqrt[7]{225}}{6}$

Step 6.12.3.10

Subtract $10 \sqrt[7]{225}$ from $45 \sqrt[7]{225}$ .

$\frac{1}{5} \cdot \frac{35 \sqrt[7]{225}}{6}$

Step 6.12.3.11

Multiply $\frac{1}{5}$ by $\frac{35 \sqrt[7]{225}}{6}$ .

$\frac{35 \sqrt[7]{225}}{5 \cdot 6}$

Step 6.12.3.12

Multiply $5$ by $6$ .

$\frac{35 \sqrt[7]{225}}{30}$

Step 6.12.3.13

Cancel the common factor of $35$ and $30$ .

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Step 6.12.3.13.1

Factor $5$ out of $35 \sqrt[7]{225}$ .

$\frac{5 (7 \sqrt[7]{225})}{30}$

Step 6.12.3.13.2

Cancel the common factors.

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Step 6.12.3.13.2.1

Factor $5$ out of $30$ .

$\frac{5 (7 \sqrt[7]{225})}{5 (6)}$

Step 6.12.3.13.2.2

Cancel the common factor.

$\frac{5 (7 \sqrt[7]{225})}{5 \cdot 6}$

Step 6.12.3.13.2.3

Rewrite the expression.

$\frac{7 \sqrt[7]{225}}{6}$

Step 7

$A r e a = \int_{\sqrt[7]{15}}^{5} 0 d x - \int_{\sqrt[7]{15}}^{5} \frac{x (15 - x^{7})}{5} d x$

Step 8

Integrate to find the area between $\sqrt[7]{15}$ and $5$ .

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Step 8.1

Combine the integrals into a single integral.

$\int_{\sqrt[7]{15}}^{5} 0 - (\frac{x (15 - x^{7})}{5}) d x$

Step 8.2

Subtract $\frac{x (15 - x^{7})}{5}$ from $0$ .

$\int_{\sqrt[7]{15}}^{5} - \frac{x (15 - x^{7})}{5} d x$

Step 8.3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int_{\sqrt[7]{15}}^{5} \frac{x (15 - x^{7})}{5} d x$

Step 8.4

Since $\frac{1}{5}$ is constant with respect to $x$ , move $\frac{1}{5}$ out of the integral.

$- (\frac{1}{5} \int_{\sqrt[7]{15}}^{5} x (15 - x^{7}) d x)$

Step 8.5

Multiply $x (15 - x^{7})$ .

$- \frac{1}{5} \int_{\sqrt[7]{15}}^{5} x \cdot 15 + x (- x^{7}) d x$

Step 8.6

Simplify.

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Step 8.6.1

Move $15$ to the left of $x$ .

$- \frac{1}{5} \int_{\sqrt[7]{15}}^{5} 15 \cdot x + x (- x^{7}) d x$

Step 8.6.2

Raise $x$ to the power of $1$ .

$- \frac{1}{5} \int_{\sqrt[7]{15}}^{5} 15 x - (x^{1} x^{7}) d x$

Step 8.6.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{5} \int_{\sqrt[7]{15}}^{5} 15 x - x^{1 + 7} d x$

Step 8.6.4

Add $1$ and $7$ .

$- \frac{1}{5} \int_{\sqrt[7]{15}}^{5} 15 x - x^{8} d x$

Step 8.7

Split the single integral into multiple integrals.

$- \frac{1}{5} (\int_{\sqrt[7]{15}}^{5} 15 x d x + \int_{\sqrt[7]{15}}^{5} - x^{8} d x)$

Step 8.8

Since $15$ is constant with respect to $x$ , move $15$ out of the integral.

$- \frac{1}{5} (15 \int_{\sqrt[7]{15}}^{5} x d x + \int_{\sqrt[7]{15}}^{5} - x^{8} d x)$

Step 8.9

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- \frac{1}{5} (15 (\frac{1}{2} x^{2}]_{\sqrt[7]{15}}^{5}) + \int_{\sqrt[7]{15}}^{5} - x^{8} d x)$

Step 8.10

Combine $\frac{1}{2}$ and $x^{2}$ .

$- \frac{1}{5} (15 (\frac{x^{2}}{2}]_{\sqrt[7]{15}}^{5}) + \int_{\sqrt[7]{15}}^{5} - x^{8} d x)$

Step 8.11

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \frac{1}{5} (15 (\frac{x^{2}}{2}]_{\sqrt[7]{15}}^{5}) - \int_{\sqrt[7]{15}}^{5} x^{8} d x)$

Step 8.12

By the Power Rule, the integral of $x^{8}$ with respect to $x$ is $\frac{1}{9} x^{9}$ .

$- \frac{1}{5} (15 (\frac{x^{2}}{2}]_{\sqrt[7]{15}}^{5}) - (\frac{1}{9} x^{9}]_{\sqrt[7]{15}}^{5}))$

Step 8.13

Simplify the answer.

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Step 8.13.1

Combine $\frac{1}{9}$ and $x^{9}$ .

$- \frac{1}{5} (15 (\frac{x^{2}}{2}]_{\sqrt[7]{15}}^{5}) - (\frac{x^{9}}{9}]_{\sqrt[7]{15}}^{5}))$

Step 8.13.2

Substitute and simplify.

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Step 8.13.2.1

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $\sqrt[7]{15}$ .

$- \frac{1}{5} (15 ((\frac{5^{2}}{2}) - \frac{{\sqrt[7]{15}}^{2}}{2}) - (\frac{x^{9}}{9}]_{\sqrt[7]{15}}^{5}))$

Step 8.13.2.2

Evaluate $\frac{x^{9}}{9}$ at $5$ and at $\sqrt[7]{15}$ .

$- \frac{1}{5} (15 (\frac{5^{2}}{2} - \frac{{\sqrt[7]{15}}^{2}}{2}) - (\frac{5^{9}}{9} - \frac{{\sqrt[7]{15}}^{9}}{9}))$

Step 8.13.2.3

Simplify.

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Step 8.13.2.3.1

Raise $5$ to the power of $2$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{{\sqrt[7]{15}}^{2}}{2}) - (\frac{5^{9}}{9} - \frac{{\sqrt[7]{15}}^{9}}{9}))$

Step 8.13.2.3.2

Rewrite ${\sqrt[7]{15}}^{2}$ as $\sqrt[7]{15^{2}}$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{15^{2}}}{2}) - (\frac{5^{9}}{9} - \frac{{\sqrt[7]{15}}^{9}}{9}))$

Step 8.13.2.3.3

Raise $15$ to the power of $2$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{5^{9}}{9} - \frac{{\sqrt[7]{15}}^{9}}{9}))$

Step 8.13.2.3.4

Raise $5$ to the power of $9$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{{\sqrt[7]{15}}^{9}}{9}))$

Step 8.13.2.3.5

Rewrite ${\sqrt[7]{15}}^{9}$ as $\sqrt[7]{15^{9}}$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{\sqrt[7]{15^{9}}}{9}))$

Step 8.13.2.3.6

Raise $15$ to the power of $9$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{\sqrt[7]{38443359375}}{9}))$

Step 8.13.3

Simplify.

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Step 8.13.3.1

Rewrite $38443359375$ as $15^{7} \cdot 225$ .

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Step 8.13.3.1.1

Factor $170859375$ out of $38443359375$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{\sqrt[7]{170859375 (225)}}{9}))$

Step 8.13.3.1.2

Rewrite $170859375$ as $15^{7}$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{\sqrt[7]{15^{7} \cdot 225}}{9}))$

Step 8.13.3.2

Pull terms out from under the radical.

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{15 \sqrt[7]{225}}{9}))$

Step 8.13.3.3

Cancel the common factor of $15$ and $9$ .

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Step 8.13.3.3.1

Factor $3$ out of $15 \sqrt[7]{225}$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{3 (5 \sqrt[7]{225})}{9}))$

Step 8.13.3.3.2

Cancel the common factors.

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Step 8.13.3.3.2.1

Factor $3$ out of $9$ .

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{3 (5 \sqrt[7]{225})}{3 (3)}))$

Step 8.13.3.3.2.2

Cancel the common factor.

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{3 (5 \sqrt[7]{225})}{3 \cdot 3}))$

Step 8.13.3.3.2.3

Rewrite the expression.

$- \frac{1}{5} (15 (\frac{25}{2} - \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{5 \sqrt[7]{225}}{3}))$

Step 8.13.4

Simplify.

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Step 8.13.4.1

Apply the distributive property.

$- \frac{1}{5} (15 (\frac{25}{2}) + 15 (- \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{5 \sqrt[7]{225}}{3}))$

Step 8.13.4.2

Multiply $15 (\frac{25}{2})$ .

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Step 8.13.4.2.1

Combine $15$ and $\frac{25}{2}$ .

$- \frac{1}{5} (\frac{15 \cdot 25}{2} + 15 (- \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{5 \sqrt[7]{225}}{3}))$

Step 8.13.4.2.2

Multiply $15$ by $25$ .

$- \frac{1}{5} (\frac{375}{2} + 15 (- \frac{\sqrt[7]{225}}{2}) - (\frac{1953125}{9} - \frac{5 \sqrt[7]{225}}{3}))$

Step 8.13.4.3

Multiply $15 (- \frac{\sqrt[7]{225}}{2})$ .

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Step 8.13.4.3.1

Multiply $- 1$ by $15$ .

$- \frac{1}{5} (\frac{375}{2} - 15 \frac{\sqrt[7]{225}}{2} - (\frac{1953125}{9} - \frac{5 \sqrt[7]{225}}{3}))$

Step 8.13.4.3.2

Combine $- 15$ and $\frac{\sqrt[7]{225}}{2}$ .

$- \frac{1}{5} (\frac{375}{2} + \frac{- 15 \sqrt[7]{225}}{2} - (\frac{1953125}{9} - \frac{5 \sqrt[7]{225}}{3}))$

Step 8.13.4.4

Apply the distributive property.

$- \frac{1}{5} (\frac{375}{2} + \frac{- 15 \sqrt[7]{225}}{2} - \frac{1953125}{9} - - \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.5

Multiply $- - \frac{5 \sqrt[7]{225}}{3}$ .

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Step 8.13.4.5.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{5} (\frac{375}{2} + \frac{- 15 \sqrt[7]{225}}{2} - \frac{1953125}{9} + 1 \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.5.2

Multiply $\frac{5 \sqrt[7]{225}}{3}$ by $1$ .

$- \frac{1}{5} (\frac{375}{2} + \frac{- 15 \sqrt[7]{225}}{2} - \frac{1953125}{9} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.6

Move the negative in front of the fraction.

$- \frac{1}{5} (\frac{375}{2} - \frac{15 \sqrt[7]{225}}{2} - \frac{1953125}{9} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.7

To write $\frac{375}{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- \frac{1}{5} (\frac{375}{2} \cdot \frac{9}{9} - \frac{1953125}{9} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.8

To write $- \frac{1953125}{9}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{5} (\frac{375}{2} \cdot \frac{9}{9} - \frac{1953125}{9} \cdot \frac{2}{2} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.9

Write each expression with a common denominator of $18$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.13.4.9.1

Multiply $\frac{375}{2}$ by $\frac{9}{9}$ .

$- \frac{1}{5} (\frac{375 \cdot 9}{2 \cdot 9} - \frac{1953125}{9} \cdot \frac{2}{2} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.9.2

Multiply $2$ by $9$ .

$- \frac{1}{5} (\frac{375 \cdot 9}{18} - \frac{1953125}{9} \cdot \frac{2}{2} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.9.3

Multiply $\frac{1953125}{9}$ by $\frac{2}{2}$ .

$- \frac{1}{5} (\frac{375 \cdot 9}{18} - \frac{1953125 \cdot 2}{9 \cdot 2} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.9.4

Multiply $9$ by $2$ .

$- \frac{1}{5} (\frac{375 \cdot 9}{18} - \frac{1953125 \cdot 2}{18} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.10

Combine the numerators over the common denominator.

$- \frac{1}{5} (\frac{375 \cdot 9 - 1953125 \cdot 2}{18} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.11

Simplify the numerator.

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Step 8.13.4.11.1

Multiply $375$ by $9$ .

$- \frac{1}{5} (\frac{3375 - 1953125 \cdot 2}{18} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.11.2

Multiply $- 1953125$ by $2$ .

$- \frac{1}{5} (\frac{3375 - 3906250}{18} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.11.3

Subtract $3906250$ from $3375$ .

$- \frac{1}{5} (\frac{- 3902875}{18} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.12

Move the negative in front of the fraction.

$- \frac{1}{5} (- \frac{3902875}{18} - \frac{15 \sqrt[7]{225}}{2} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.13

To write $- \frac{15 \sqrt[7]{225}}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{1}{5} (- \frac{3902875}{18} - \frac{15 \sqrt[7]{225}}{2} \cdot \frac{3}{3} + \frac{5 \sqrt[7]{225}}{3})$

Step 8.13.4.14

To write $\frac{5 \sqrt[7]{225}}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{5} (- \frac{3902875}{18} - \frac{15 \sqrt[7]{225}}{2} \cdot \frac{3}{3} + \frac{5 \sqrt[7]{225}}{3} \cdot \frac{2}{2})$

Step 8.13.4.15

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 8.13.4.15.1

Multiply $\frac{15 \sqrt[7]{225}}{2}$ by $\frac{3}{3}$ .

$- \frac{1}{5} (- \frac{3902875}{18} - \frac{15 \sqrt[7]{225} \cdot 3}{2 \cdot 3} + \frac{5 \sqrt[7]{225}}{3} \cdot \frac{2}{2})$

Step 8.13.4.15.2

Multiply $2$ by $3$ .

$- \frac{1}{5} (- \frac{3902875}{18} - \frac{15 \sqrt[7]{225} \cdot 3}{6} + \frac{5 \sqrt[7]{225}}{3} \cdot \frac{2}{2})$

Step 8.13.4.15.3

Multiply $\frac{5 \sqrt[7]{225}}{3}$ by $\frac{2}{2}$ .

$- \frac{1}{5} (- \frac{3902875}{18} - \frac{15 \sqrt[7]{225} \cdot 3}{6} + \frac{5 \sqrt[7]{225} \cdot 2}{3 \cdot 2})$

Step 8.13.4.15.4

Multiply $3$ by $2$ .

$- \frac{1}{5} (- \frac{3902875}{18} - \frac{15 \sqrt[7]{225} \cdot 3}{6} + \frac{5 \sqrt[7]{225} \cdot 2}{6})$

Step 8.13.4.16

Combine the numerators over the common denominator.

$- \frac{1}{5} (- \frac{3902875}{18} + \frac{- 15 \sqrt[7]{225} \cdot 3 + 5 \sqrt[7]{225} \cdot 2}{6})$

Step 8.13.4.17

Multiply $3$ by $- 15$ .

$- \frac{1}{5} (- \frac{3902875}{18} + \frac{- 45 \sqrt[7]{225} + 5 \sqrt[7]{225} \cdot 2}{6})$

Step 8.13.4.18

Multiply $2$ by $5$ .

$- \frac{1}{5} (- \frac{3902875}{18} + \frac{- 45 \sqrt[7]{225} + 10 \sqrt[7]{225}}{6})$

Step 8.13.4.19

Add $- 45 \sqrt[7]{225}$ and $10 \sqrt[7]{225}$ .

$- \frac{1}{5} (- \frac{3902875}{18} + \frac{- 35 \sqrt[7]{225}}{6})$

Step 8.13.4.20

Move the negative in front of the fraction.

$- \frac{1}{5} (- \frac{3902875}{18} - \frac{35 \sqrt[7]{225}}{6})$

Step 8.13.4.21

Apply the distributive property.

$- \frac{1}{5} (- \frac{3902875}{18}) - \frac{1}{5} (- \frac{35 \sqrt[7]{225}}{6})$

Step 8.13.4.22

Cancel the common factor of $5$ .

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Step 8.13.4.22.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$\frac{- 1}{5} (- \frac{3902875}{18}) - \frac{1}{5} (- \frac{35 \sqrt[7]{225}}{6})$

Step 8.13.4.22.2

Move the leading negative in $- \frac{3902875}{18}$ into the numerator.

$\frac{- 1}{5} \cdot \frac{- 3902875}{18} - \frac{1}{5} (- \frac{35 \sqrt[7]{225}}{6})$

Step 8.13.4.22.3

Factor $5$ out of $- 3902875$ .

$\frac{- 1}{5} \cdot \frac{5 (- 780575)}{18} - \frac{1}{5} (- \frac{35 \sqrt[7]{225}}{6})$

Step 8.13.4.22.4

Cancel the common factor.

$\frac{- 1}{5} \cdot \frac{5 \cdot - 780575}{18} - \frac{1}{5} (- \frac{35 \sqrt[7]{225}}{6})$

Step 8.13.4.22.5

Rewrite the expression.

$- \frac{- 780575}{18} - \frac{1}{5} (- \frac{35 \sqrt[7]{225}}{6})$

Step 8.13.4.23

Cancel the common factor of $5$ .

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Step 8.13.4.23.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$- \frac{- 780575}{18} + \frac{- 1}{5} (- \frac{35 \sqrt[7]{225}}{6})$

Step 8.13.4.23.2

Move the leading negative in $- \frac{35 \sqrt[7]{225}}{6}$ into the numerator.

$- \frac{- 780575}{18} + \frac{- 1}{5} \cdot \frac{- 35 \sqrt[7]{225}}{6}$

Step 8.13.4.23.3

Factor $5$ out of $- 35 \sqrt[7]{225}$ .

$- \frac{- 780575}{18} + \frac{- 1}{5} \cdot \frac{5 (- 7 \sqrt[7]{225})}{6}$

Step 8.13.4.23.4

Cancel the common factor.

$- \frac{- 780575}{18} + \frac{- 1}{5} \cdot \frac{5 (- 7 \sqrt[7]{225})}{6}$

Step 8.13.4.23.5

Rewrite the expression.

$- \frac{- 780575}{18} - \frac{- 7 \sqrt[7]{225}}{6}$

Step 8.13.4.24

Move the negative in front of the fraction.

$- - \frac{780575}{18} - \frac{- 7 \sqrt[7]{225}}{6}$

Step 8.13.4.25

Multiply $- - \frac{780575}{18}$ .

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Step 8.13.4.25.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{780575}{18}) - \frac{- 7 \sqrt[7]{225}}{6}$

Step 8.13.4.25.2

Multiply $\frac{780575}{18}$ by $1$ .

$\frac{780575}{18} - \frac{- 7 \sqrt[7]{225}}{6}$

Step 8.13.4.26

Move the negative in front of the fraction.

$\frac{780575}{18} - - \frac{7 \sqrt[7]{225}}{6}$

Step 8.13.4.27

Multiply $- - \frac{7 \sqrt[7]{225}}{6}$ .

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Step 8.13.4.27.1

Multiply $- 1$ by $- 1$ .

$\frac{780575}{18} + 1 \frac{7 \sqrt[7]{225}}{6}$

Step 8.13.4.27.2

Multiply $\frac{7 \sqrt[7]{225}}{6}$ by $1$ .

$\frac{780575}{18} + \frac{7 \sqrt[7]{225}}{6}$

Step 9

Add the areas $\frac{781925}{18} + \frac{7 \sqrt[7]{225}}{6} + \frac{780575}{18} + \frac{7 \sqrt[7]{225}}{6}$ .

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Step 9.1

Simplify terms.

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Step 9.1.1

Combine the numerators over the common denominator.

$\frac{781925 + 780575}{18} + \frac{7 \sqrt[7]{225} + 7 \sqrt[7]{225}}{6}$

Step 9.1.2

Add $781925$ and $780575$ .

$\frac{1562500}{18} + \frac{7 \sqrt[7]{225} + 7 \sqrt[7]{225}}{6}$

Step 9.1.3

Add $7 \sqrt[7]{225}$ and $7 \sqrt[7]{225}$ .

$\frac{1562500}{18} + \frac{14 \sqrt[7]{225}}{6}$

Step 9.2

Simplify each term.

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Step 9.2.1

Cancel the common factor of $1562500$ and $18$ .

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Step 9.2.1.1

Factor $2$ out of $1562500$ .

$\frac{2 (781250)}{18} + \frac{14 \sqrt[7]{225}}{6}$

Step 9.2.1.2

Cancel the common factors.

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Step 9.2.1.2.1

Factor $2$ out of $18$ .

$\frac{2 \cdot 781250}{2 \cdot 9} + \frac{14 \sqrt[7]{225}}{6}$

Step 9.2.1.2.2

Cancel the common factor.

$\frac{2 \cdot 781250}{2 \cdot 9} + \frac{14 \sqrt[7]{225}}{6}$

Step 9.2.1.2.3

Rewrite the expression.

$\frac{781250}{9} + \frac{14 \sqrt[7]{225}}{6}$

Step 9.2.2

Cancel the common factor of $14$ and $6$ .

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Step 9.2.2.1

Factor $2$ out of $14 \sqrt[7]{225}$ .

$\frac{781250}{9} + \frac{2 (7 \sqrt[7]{225})}{6}$

Step 9.2.2.2

Cancel the common factors.

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Step 9.2.2.2.1

Factor $2$ out of $6$ .

$\frac{781250}{9} + \frac{2 (7 \sqrt[7]{225})}{2 (3)}$

Step 9.2.2.2.2

Cancel the common factor.

$\frac{781250}{9} + \frac{2 (7 \sqrt[7]{225})}{2 \cdot 3}$

Step 9.2.2.2.3

Rewrite the expression.

$\frac{781250}{9} + \frac{7 \sqrt[7]{225}}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{781250}{9} + \frac{7 \sqrt[7]{225}}{3}$

Decimal Form:

$86810.61383547 \dots$

Step 11