Calculus Examples

$- 5 {(x^{2} - 24 x + 80)}^{\frac{4}{5}}$

Step 1

Find the first derivative.

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Step 1.1

Find the first derivative.

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Step 1.1.1

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 {(x^{2} - 24 x + 80)}^{\frac{4}{5}}$ with respect to $x$ is $- 5 \frac{d}{d x} [{(x^{2} - 24 x + 80)}^{\frac{4}{5}}]$ .

$- 5 \frac{d}{d x} [{(x^{2} - 24 x + 80)}^{\frac{4}{5}}]$

Step 1.1.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{4}{5}}$ and $g (x) = x^{2} - 24 x + 80$ .

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $x^{2} - 24 x + 80$ .

$- 5 (\frac{d}{d u} [u^{\frac{4}{5}}] \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{4}{5}$ .

$- 5 (\frac{4}{5} u^{\frac{4}{5} - 1} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.2.3

Replace all occurrences of $u$ with $x^{2} - 24 x + 80$ .

$- 5 (\frac{4}{5} {(x^{2} - 24 x + 80)}^{\frac{4}{5} - 1} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 5 (\frac{4}{5} {(x^{2} - 24 x + 80)}^{\frac{4}{5} - 1 \cdot \frac{5}{5}} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.4

Combine $- 1$ and $\frac{5}{5}$ .

$- 5 (\frac{4}{5} {(x^{2} - 24 x + 80)}^{\frac{4}{5} + \frac{- 1 \cdot 5}{5}} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.5

Combine the numerators over the common denominator.

$- 5 (\frac{4}{5} {(x^{2} - 24 x + 80)}^{\frac{4 - 1 \cdot 5}{5}} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $5$ .

$- 5 (\frac{4}{5} {(x^{2} - 24 x + 80)}^{\frac{4 - 5}{5}} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.6.2

Subtract $5$ from $4$ .

$- 5 (\frac{4}{5} {(x^{2} - 24 x + 80)}^{\frac{- 1}{5}} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.7

Move the negative in front of the fraction.

$- 5 (\frac{4}{5} {(x^{2} - 24 x + 80)}^{- \frac{1}{5}} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.8

Combine $\frac{4}{5}$ and ${(x^{2} - 24 x + 80)}^{- \frac{1}{5}}$ .

$- 5 (\frac{4 {(x^{2} - 24 x + 80)}^{- \frac{1}{5}}}{5} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.9

Move ${(x^{2} - 24 x + 80)}^{- \frac{1}{5}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 5 (\frac{4}{5 {(x^{2} - 24 x + 80)}^{\frac{1}{5}}} \frac{d}{d x} [x^{2} - 24 x + 80])$

Step 1.1.10

Combine $\frac{4}{5 {(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$ and $- 5$ .

$\frac{4 \cdot - 5}{5 {(x^{2} - 24 x + 80)}^{\frac{1}{5}}} \frac{d}{d x} [x^{2} - 24 x + 80]$

Step 1.1.11

Multiply $4$ by $- 5$ .

$\frac{- 20}{5 {(x^{2} - 24 x + 80)}^{\frac{1}{5}}} \frac{d}{d x} [x^{2} - 24 x + 80]$

Step 1.1.12

Factor $5$ out of $- 20$ .

$\frac{5 \cdot - 4}{5 {(x^{2} - 24 x + 80)}^{\frac{1}{5}}} \frac{d}{d x} [x^{2} - 24 x + 80]$

Step 1.1.13

Cancel the common factors.

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Step 1.1.13.1

Factor $5$ out of $5 {(x^{2} - 24 x + 80)}^{\frac{1}{5}}$ .

$\frac{5 \cdot - 4}{5 ({(x^{2} - 24 x + 80)}^{\frac{1}{5}})} \frac{d}{d x} [x^{2} - 24 x + 80]$

Step 1.1.13.2

Cancel the common factor.

$\frac{5 \cdot - 4}{5 {(x^{2} - 24 x + 80)}^{\frac{1}{5}}} \frac{d}{d x} [x^{2} - 24 x + 80]$

Step 1.1.13.3

Rewrite the expression.

$\frac{- 4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} \frac{d}{d x} [x^{2} - 24 x + 80]$

Step 1.1.14

Move the negative in front of the fraction.

$- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} \frac{d}{d x} [x^{2} - 24 x + 80]$

Step 1.1.15

By the Sum Rule, the derivative of $x^{2} - 24 x + 80$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [80]$ .

$- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [80])$

Step 1.1.16

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} (2 x + \frac{d}{d x} [- 24 x] + \frac{d}{d x} [80])$

Step 1.1.17

Since $- 24$ is constant with respect to $x$ , the derivative of $- 24 x$ with respect to $x$ is $- 24 \frac{d}{d x} [x]$ .

$- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} (2 x - 24 \frac{d}{d x} [x] + \frac{d}{d x} [80])$

Step 1.1.18

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} (2 x - 24 \cdot 1 + \frac{d}{d x} [80])$

Step 1.1.19

Multiply $- 24$ by $1$ .

$- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} (2 x - 24 + \frac{d}{d x} [80])$

Step 1.1.20

Since $80$ is constant with respect to $x$ , the derivative of $80$ with respect to $x$ is $0$ .

$- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} (2 x - 24 + 0)$

Step 1.1.21

Add $2 x - 24$ and $0$ .

$- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} (2 x - 24)$

Step 1.1.22

Simplify.

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Step 1.1.22.1

Reorder the factors of $- \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} (2 x - 24)$ .

$- (2 x - 24) \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.2

Apply the distributive property.

$(- (2 x) - - 24) \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.3

Multiply $2$ by $- 1$ .

$(- 2 x - - 24) \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.4

Multiply $- 1$ by $- 24$ .

$(- 2 x + 24) \frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.5

Multiply $- 2 x + 24$ by $\frac{4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$ .

$\frac{(- 2 x + 24) \cdot 4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.6

Simplify the numerator.

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Step 1.1.22.6.1

Factor $2$ out of $- 2 x + 24$ .

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Step 1.1.22.6.1.1

Factor $2$ out of $- 2 x$ .

$\frac{(2 (- x) + 24) \cdot 4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.6.1.2

Factor $2$ out of $24$ .

$\frac{(2 (- x) + 2 (12)) \cdot 4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.6.1.3

Factor $2$ out of $2 (- x) + 2 (12)$ .

$\frac{2 (- x + 12) \cdot 4}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.6.2

Multiply $4$ by $2$ .

$\frac{8 (- x + 12)}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.7

Factor $- 1$ out of $- x$ .

$\frac{8 (- (x) + 12)}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.8

Rewrite $12$ as $- 1 (- 12)$ .

$\frac{8 (- (x) - 1 (- 12))}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.9

Factor $- 1$ out of $- (x) - 1 (- 12)$ .

$\frac{8 (- (x - 12))}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.10

Rewrite $- (x - 12)$ as $- 1 (x - 12)$ .

$\frac{8 (- 1 (x - 12))}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.1.22.11

Move the negative in front of the fraction.

$f' (x) = - \frac{8 (x - 12)}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $- \frac{8 (x - 12)}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$ .

$- \frac{8 (x - 12)}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $- \frac{8 (x - 12)}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$- \frac{8 (x - 12)}{{(x^{2} - 24 x + 80)}^{\frac{1}{5}}} = 0$

Step 2.2

Set the numerator equal to zero.

$8 (x - 12) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $8 (x - 12) = 0$ by $8$ and simplify.

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Step 2.3.1.1

Divide each term in $8 (x - 12) = 0$ by $8$ .

$\frac{8 (x - 12)}{8} = \frac{0}{8}$

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $8$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

$\frac{8 (x - 12)}{8} = \frac{0}{8}$

Step 2.3.1.2.1.2

Divide $x - 12$ by $1$ .

$x - 12 = \frac{0}{8}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $8$ .

$x - 12 = 0$

Step 2.3.2

Add $12$ to both sides of the equation.

$x = 12$

Step 3

Find the values where the derivative is undefined.

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Step 3.1

Convert expressions with fractional exponents to radicals.

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Step 3.1.1

Apply the rule $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$ to rewrite the exponentiation as a radical.

$- \frac{8 (x - 12)}{\sqrt[5]{{(x^{2} - 24 x + 80)}^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$- \frac{8 (x - 12)}{\sqrt[5]{x^{2} - 24 x + 80}}$

Step 3.2

Set the denominator in $\frac{8 (x - 12)}{\sqrt[5]{x^{2} - 24 x + 80}}$ equal to $0$ to find where the expression is undefined.

$\sqrt[5]{x^{2} - 24 x + 80} = 0$

Step 3.3

Solve for $x$ .

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Step 3.3.1

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $5$ .

${\sqrt[5]{x^{2} - 24 x + 80}}^{5} = 0^{5}$

Step 3.3.2

Simplify each side of the equation.

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Step 3.3.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x^{2} - 24 x + 80}$ as ${(x^{2} - 24 x + 80)}^{\frac{1}{5}}$ .

${({(x^{2} - 24 x + 80)}^{\frac{1}{5}})}^{5} = 0^{5}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${({(x^{2} - 24 x + 80)}^{\frac{1}{5}})}^{5}$ .

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Step 3.3.2.2.1.1

Multiply the exponents in ${({(x^{2} - 24 x + 80)}^{\frac{1}{5}})}^{5}$ .

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Step 3.3.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(x^{2} - 24 x + 80)}^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 3.3.2.2.1.1.2

Cancel the common factor of $5$ .

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Step 3.3.2.2.1.1.2.1

Cancel the common factor.

${(x^{2} - 24 x + 80)}^{\frac{1}{5} \cdot 5} = 0^{5}$

Step 3.3.2.2.1.1.2.2

Rewrite the expression.

${(x^{2} - 24 x + 80)}^{1} = 0^{5}$

Step 3.3.2.2.1.2

Simplify.

$x^{2} - 24 x + 80 = 0^{5}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

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

$x^{2} - 24 x + 80 = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Factor $x^{2} - 24 x + 80$ using the AC method.

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Step 3.3.3.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $80$ and whose sum is $- 24$ .

$- 20, - 4$

Step 3.3.3.1.2

Write the factored form using these integers.

$(x - 20) (x - 4) = 0$

Step 3.3.3.2

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

$x - 20 = 0$

$x - 4 = 0$

Step 3.3.3.3

Set $x - 20$ equal to $0$ and solve for $x$ .

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Step 3.3.3.3.1

Set $x - 20$ equal to $0$ .

$x - 20 = 0$

Step 3.3.3.3.2

Add $20$ to both sides of the equation.

$x = 20$

Step 3.3.3.4

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 3.3.3.4.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 3.3.3.4.2

Add $4$ to both sides of the equation.

$x = 4$

Step 3.3.3.5

The final solution is all the values that make $(x - 20) (x - 4) = 0$ true.

$x = 20, 4$

Step 3.4

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = 4, x = 20$

Step 4

Evaluate $- 5 {(x^{2} - 24 x + 80)}^{\frac{4}{5}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = 12$ .

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Step 4.1.1

Substitute $12$ for $x$ .

$- 5 {({(12)}^{2} - 24 \cdot 12 + 80)}^{\frac{4}{5}}$

Step 4.1.2

Simplify.

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Step 4.1.2.1

Simplify each term.

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Step 4.1.2.1.1

Raise $12$ to the power of $2$ .

$- 5 {(144 - 24 \cdot 12 + 80)}^{\frac{4}{5}}$

Step 4.1.2.1.2

Multiply $- 24$ by $12$ .

$- 5 {(144 - 288 + 80)}^{\frac{4}{5}}$

Step 4.1.2.2

Simplify by adding and subtracting.

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Step 4.1.2.2.1

Subtract $288$ from $144$ .

$- 5 {(- 144 + 80)}^{\frac{4}{5}}$

Step 4.1.2.2.2

Add $- 144$ and $80$ .

$- 5 {(- 64)}^{\frac{4}{5}}$

Step 4.2

Evaluate at $x = 4$ .

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Step 4.2.1

Substitute $4$ for $x$ .

$- 5 {({(4)}^{2} - 24 \cdot 4 + 80)}^{\frac{4}{5}}$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify each term.

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Step 4.2.2.1.1

Raise $4$ to the power of $2$ .

$- 5 {(16 - 24 \cdot 4 + 80)}^{\frac{4}{5}}$

Step 4.2.2.1.2

Multiply $- 24$ by $4$ .

$- 5 {(16 - 96 + 80)}^{\frac{4}{5}}$

Step 4.2.2.2

Reduce the expression by cancelling the common factors.

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Step 4.2.2.2.1

Subtract $96$ from $16$ .

$- 5 {(- 80 + 80)}^{\frac{4}{5}}$

Step 4.2.2.2.2

Simplify the expression.

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Step 4.2.2.2.2.1

Add $- 80$ and $80$ .

$- 5 \cdot 0^{\frac{4}{5}}$

Step 4.2.2.2.2.2

Rewrite $0$ as $0^{5}$ .

$- 5 \cdot {(0^{5})}^{\frac{4}{5}}$

Step 4.2.2.2.2.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 5 \cdot 0^{5 (\frac{4}{5})}$

Step 4.2.2.2.3

Cancel the common factor of $5$ .

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Step 4.2.2.2.3.1

Cancel the common factor.

$- 5 \cdot 0^{5 (\frac{4}{5})}$

Step 4.2.2.2.3.2

Rewrite the expression.

$- 5 \cdot 0^{4}$

Step 4.2.2.2.4

Simplify the expression.

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Step 4.2.2.2.4.1

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

$- 5 \cdot 0$

Step 4.2.2.2.4.2

Multiply $- 5$ by $0$ .

$0$

Step 4.3

Evaluate at $x = 20$ .

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Step 4.3.1

Substitute $20$ for $x$ .

$- 5 {({(20)}^{2} - 24 \cdot 20 + 80)}^{\frac{4}{5}}$

Step 4.3.2

Simplify.

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Step 4.3.2.1

Simplify each term.

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Step 4.3.2.1.1

Raise $20$ to the power of $2$ .

$- 5 {(400 - 24 \cdot 20 + 80)}^{\frac{4}{5}}$

Step 4.3.2.1.2

Multiply $- 24$ by $20$ .

$- 5 {(400 - 480 + 80)}^{\frac{4}{5}}$

Step 4.3.2.2

Reduce the expression by cancelling the common factors.

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Step 4.3.2.2.1

Subtract $480$ from $400$ .

$- 5 {(- 80 + 80)}^{\frac{4}{5}}$

Step 4.3.2.2.2

Simplify the expression.

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Step 4.3.2.2.2.1

Add $- 80$ and $80$ .

$- 5 \cdot 0^{\frac{4}{5}}$

Step 4.3.2.2.2.2

Rewrite $0$ as $0^{5}$ .

$- 5 \cdot {(0^{5})}^{\frac{4}{5}}$

Step 4.3.2.2.2.3

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$- 5 \cdot 0^{5 (\frac{4}{5})}$

Step 4.3.2.2.3

Cancel the common factor of $5$ .

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Step 4.3.2.2.3.1

Cancel the common factor.

$- 5 \cdot 0^{5 (\frac{4}{5})}$

Step 4.3.2.2.3.2

Rewrite the expression.

$- 5 \cdot 0^{4}$

Step 4.3.2.2.4

Simplify the expression.

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Step 4.3.2.2.4.1

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

$- 5 \cdot 0$

Step 4.3.2.2.4.2

Multiply $- 5$ by $0$ .

$0$

Step 4.4

List all of the points.

$(12, - 5 {(- 64)}^{\frac{4}{5}}), (4, 0), (20, 0)$

Step 5