Calculus Examples

$f (x) = \frac{1}{3} x^{3} - x^{\frac{2}{3}}$

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

By the Sum Rule, the derivative of $\frac{1}{3} x^{3} - x^{\frac{2}{3}}$ with respect to $x$ is $\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- x^{\frac{2}{3}}]$ .

$\frac{d}{d x} [\frac{1}{3} x^{3}] + \frac{d}{d x} [- x^{\frac{2}{3}}]$

Step 1.1.2

Evaluate $\frac{d}{d x} [\frac{1}{3} x^{3}]$ .

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Step 1.1.2.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{1}{3} x^{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x^{3}]$ .

$\frac{1}{3} \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{\frac{2}{3}}]$

Step 1.1.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$\frac{1}{3} (3 x^{2}) + \frac{d}{d x} [- x^{\frac{2}{3}}]$

Step 1.1.2.3

Combine $3$ and $\frac{1}{3}$ .

$\frac{3}{3} x^{2} + \frac{d}{d x} [- x^{\frac{2}{3}}]$

Step 1.1.2.4

Combine $\frac{3}{3}$ and $x^{2}$ .

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- x^{\frac{2}{3}}]$

Step 1.1.2.5

Cancel the common factor of $3$ .

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Step 1.1.2.5.1

Cancel the common factor.

$\frac{3 x^{2}}{3} + \frac{d}{d x} [- x^{\frac{2}{3}}]$

Step 1.1.2.5.2

Divide $x^{2}$ by $1$ .

$x^{2} + \frac{d}{d x} [- x^{\frac{2}{3}}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x^{\frac{2}{3}}]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{2}{3}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{2}{3}}]$ .

$x^{2} - \frac{d}{d x} [x^{\frac{2}{3}}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{2}{3}$ .

$x^{2} - (\frac{2}{3} x^{\frac{2}{3} - 1})$

Step 1.1.3.3

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$x^{2} - (\frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}})$

Step 1.1.3.4

Combine $- 1$ and $\frac{3}{3}$ .

$x^{2} - (\frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.1.3.5

Combine the numerators over the common denominator.

$x^{2} - (\frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}})$

Step 1.1.3.6

Simplify the numerator.

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Step 1.1.3.6.1

Multiply $- 1$ by $3$ .

$x^{2} - (\frac{2}{3} x^{\frac{2 - 3}{3}})$

Step 1.1.3.6.2

Subtract $3$ from $2$ .

$x^{2} - (\frac{2}{3} x^{\frac{- 1}{3}})$

Step 1.1.3.7

Move the negative in front of the fraction.

$x^{2} - (\frac{2}{3} x^{- \frac{1}{3}})$

Step 1.1.3.8

Combine $\frac{2}{3}$ and $x^{- \frac{1}{3}}$ .

$x^{2} - \frac{2 x^{- \frac{1}{3}}}{3}$

Step 1.1.3.9

Move $x^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = x^{2} - \frac{2}{3 x^{\frac{1}{3}}}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $x^{2} - \frac{2}{3 x^{\frac{1}{3}}}$ .

$x^{2} - \frac{2}{3 x^{\frac{1}{3}}}$

Step 2

Set the first derivative equal to $0$ then solve the equation $x^{2} - \frac{2}{3 x^{\frac{1}{3}}} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$x^{2} - \frac{2}{3 x^{\frac{1}{3}}} = 0$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, 3 x^{\frac{1}{3}}, 1$

Step 2.2.2

The LCM of one and any expression is the expression.

$3 x^{\frac{1}{3}}$

Step 2.3

Multiply each term in $x^{2} - \frac{2}{3 x^{\frac{1}{3}}} = 0$ by $3 x^{\frac{1}{3}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $x^{2} - \frac{2}{3 x^{\frac{1}{3}}} = 0$ by $3 x^{\frac{1}{3}}$ .

$x^{2} (3 x^{\frac{1}{3}}) - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Simplify each term.

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Step 2.3.2.1.1

Rewrite using the commutative property of multiplication.

$3 x^{2} x^{\frac{1}{3}} - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2

Multiply $x^{2}$ by $x^{\frac{1}{3}}$ by adding the exponents.

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Step 2.3.2.1.2.1

Move $x^{\frac{1}{3}}$ .

$3 (x^{\frac{1}{3}} x^{2}) - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2.2

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

$3 x^{\frac{1}{3} + 2} - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$3 x^{\frac{1}{3} + 2 \cdot \frac{3}{3}} - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2.4

Combine $2$ and $\frac{3}{3}$ .

$3 x^{\frac{1}{3} + \frac{2 \cdot 3}{3}} - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2.5

Combine the numerators over the common denominator.

$3 x^{\frac{1 + 2 \cdot 3}{3}} - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2.6

Simplify the numerator.

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Step 2.3.2.1.2.6.1

Multiply $2$ by $3$ .

$3 x^{\frac{1 + 6}{3}} - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2.6.2

Add $1$ and $6$ .

$3 x^{\frac{7}{3}} - \frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.3

Cancel the common factor of $3 x^{\frac{1}{3}}$ .

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Step 2.3.2.1.3.1

Move the leading negative in $- \frac{2}{3 x^{\frac{1}{3}}}$ into the numerator.

$3 x^{\frac{7}{3}} + \frac{- 2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.3.2

Cancel the common factor.

$3 x^{\frac{7}{3}} + \frac{- 2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.3.3

Rewrite the expression.

$3 x^{\frac{7}{3}} - 2 = 0 (3 x^{\frac{1}{3}})$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Multiply $0 (3 x^{\frac{1}{3}})$ .

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Step 2.3.3.1.1

Multiply $3$ by $0$ .

$3 x^{\frac{7}{3}} - 2 = 0 x^{\frac{1}{3}}$

Step 2.3.3.1.2

Multiply $0$ by $x^{\frac{1}{3}}$ .

$3 x^{\frac{7}{3}} - 2 = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Add $2$ to both sides of the equation.

$3 x^{\frac{7}{3}} = 2$

Step 2.4.2

Raise each side of the equation to the power of $\frac{3}{7}$ to eliminate the fractional exponent on the left side.

${(3 x^{\frac{7}{3}})}^{\frac{3}{7}} = 2^{\frac{3}{7}}$

Step 2.4.3

Simplify the left side.

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Step 2.4.3.1

Simplify ${(3 x^{\frac{7}{3}})}^{\frac{3}{7}}$ .

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Step 2.4.3.1.1

Apply the product rule to $3 x^{\frac{7}{3}}$ .

$3^{\frac{3}{7}} {(x^{\frac{7}{3}})}^{\frac{3}{7}} = 2^{\frac{3}{7}}$

Step 2.4.3.1.2

Multiply the exponents in ${(x^{\frac{7}{3}})}^{\frac{3}{7}}$ .

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Step 2.4.3.1.2.1

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

$3^{\frac{3}{7}} x^{\frac{7}{3} \cdot \frac{3}{7}} = 2^{\frac{3}{7}}$

Step 2.4.3.1.2.2

Cancel the common factor of $7$ .

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Step 2.4.3.1.2.2.1

Cancel the common factor.

$3^{\frac{3}{7}} x^{\frac{7}{3} \cdot \frac{3}{7}} = 2^{\frac{3}{7}}$

Step 2.4.3.1.2.2.2

Rewrite the expression.

$3^{\frac{3}{7}} x^{\frac{1}{3} \cdot 3} = 2^{\frac{3}{7}}$

Step 2.4.3.1.2.3

Cancel the common factor of $3$ .

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Step 2.4.3.1.2.3.1

Cancel the common factor.

$3^{\frac{3}{7}} x^{\frac{1}{3} \cdot 3} = 2^{\frac{3}{7}}$

Step 2.4.3.1.2.3.2

Rewrite the expression.

$3^{\frac{3}{7}} x^{1} = 2^{\frac{3}{7}}$

Step 2.4.3.1.3

Simplify.

$3^{\frac{3}{7}} x = 2^{\frac{3}{7}}$

Step 2.4.3.1.4

Reorder factors in $3^{\frac{3}{7}} x$ .

$x \cdot 3^{\frac{3}{7}} = 2^{\frac{3}{7}}$

Step 2.4.4

Divide each term in $x \cdot 3^{\frac{3}{7}} = 2^{\frac{3}{7}}$ by $3^{\frac{3}{7}}$ and simplify.

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Step 2.4.4.1

Divide each term in $x \cdot 3^{\frac{3}{7}} = 2^{\frac{3}{7}}$ by $3^{\frac{3}{7}}$ .

$\frac{x \cdot 3^{\frac{3}{7}}}{3^{\frac{3}{7}}} = \frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}}$

Step 2.4.4.2

Simplify the left side.

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Step 2.4.4.2.1

Cancel the common factor.

$\frac{x \cdot 3^{\frac{3}{7}}}{3^{\frac{3}{7}}} = \frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}}$

Step 2.4.4.2.2

Divide $x$ by $1$ .

$x = \frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}}$

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.

$x^{2} - \frac{2}{3 \sqrt[3]{x^{1}}}$

Step 3.1.2

Anything raised to $1$ is the base itself.

$x^{2} - \frac{2}{3 \sqrt[3]{x}}$

Step 3.2

Set the denominator in $\frac{2}{3 \sqrt[3]{x}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{x} = 0$

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, cube both sides of the equation.

${(3 \sqrt[3]{x})}^{3} = 0^{3}$

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 $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

${(3 x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(3 x^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $3 x^{\frac{1}{3}}$ .

$3^{3} {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.2

Raise $3$ to the power of $3$ .

$27 {(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.3.1

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

$27 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2

Cancel the common factor of $3$ .

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Step 3.3.2.2.1.3.2.1

Cancel the common factor.

$27 x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$27 x^{1} = 0^{3}$

Step 3.3.2.2.1.4

Simplify.

$27 x = 0^{3}$

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 $0$ .

$27 x = 0$

Step 3.3.3

Divide each term in $27 x = 0$ by $27$ and simplify.

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Step 3.3.3.1

Divide each term in $27 x = 0$ by $27$ .

$\frac{27 x}{27} = \frac{0}{27}$

Step 3.3.3.2

Simplify the left side.

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Step 3.3.3.2.1

Cancel the common factor of $27$ .

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Step 3.3.3.2.1.1

Cancel the common factor.

$\frac{27 x}{27} = \frac{0}{27}$

Step 3.3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{27}$

Step 3.3.3.3

Simplify the right side.

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Step 3.3.3.3.1

Divide $0$ by $27$ .

$x = 0$

Step 4

Evaluate $\frac{1}{3} x^{3} - x^{\frac{2}{3}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}}$ .

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Step 4.1.1

Substitute $\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}}$ for $x$ .

$\frac{1}{3} \cdot {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{3} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

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

Apply the product rule to $\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}}$ .

$\frac{1}{3} \cdot \frac{{(2^{\frac{3}{7}})}^{3}}{{(3^{\frac{3}{7}})}^{3}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.2

Combine.

$\frac{1 {(2^{\frac{3}{7}})}^{3}}{3 {(3^{\frac{3}{7}})}^{3}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.3

Multiply ${(2^{\frac{3}{7}})}^{3}$ by $1$ .

$\frac{{(2^{\frac{3}{7}})}^{3}}{3 {(3^{\frac{3}{7}})}^{3}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.4

Multiply the exponents in ${(3^{\frac{3}{7}})}^{3}$ .

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Step 4.1.2.1.4.1

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

$\frac{{(2^{\frac{3}{7}})}^{3}}{3 \cdot 3^{\frac{3}{7} \cdot 3}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.4.2

Multiply $\frac{3}{7} \cdot 3$ .

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Step 4.1.2.1.4.2.1

Combine $\frac{3}{7}$ and $3$ .

$\frac{{(2^{\frac{3}{7}})}^{3}}{3 \cdot 3^{\frac{3 \cdot 3}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.4.2.2

Multiply $3$ by $3$ .

$\frac{{(2^{\frac{3}{7}})}^{3}}{3 \cdot 3^{\frac{9}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.5

Multiply the exponents in ${(2^{\frac{3}{7}})}^{3}$ .

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Step 4.1.2.1.5.1

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

$\frac{2^{\frac{3}{7} \cdot 3}}{3 \cdot 3^{\frac{9}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.5.2

Multiply $\frac{3}{7} \cdot 3$ .

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Step 4.1.2.1.5.2.1

Combine $\frac{3}{7}$ and $3$ .

$\frac{2^{\frac{3 \cdot 3}{7}}}{3 \cdot 3^{\frac{9}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.5.2.2

Multiply $3$ by $3$ .

$\frac{2^{\frac{9}{7}}}{3 \cdot 3^{\frac{9}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.6

Multiply $3$ by $3^{\frac{9}{7}}$ by adding the exponents.

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Step 4.1.2.1.6.1

Multiply $3$ by $3^{\frac{9}{7}}$ .

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Step 4.1.2.1.6.1.1

Raise $3$ to the power of $1$ .

$\frac{2^{\frac{9}{7}}}{3^{1} \cdot 3^{\frac{9}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.6.1.2

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

$\frac{2^{\frac{9}{7}}}{3^{1 + \frac{9}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.6.2

Write $1$ as a fraction with a common denominator.

$\frac{2^{\frac{9}{7}}}{3^{\frac{7}{7} + \frac{9}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.6.3

Combine the numerators over the common denominator.

$\frac{2^{\frac{9}{7}}}{3^{\frac{7 + 9}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.6.4

Add $7$ and $9$ .

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - {(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}})}^{\frac{2}{3}}$

Step 4.1.2.1.7

Apply the product rule to $\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}}$ .

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{{(2^{\frac{3}{7}})}^{\frac{2}{3}}}{{(3^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 4.1.2.1.8

Multiply the exponents in ${(2^{\frac{3}{7}})}^{\frac{2}{3}}$ .

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Step 4.1.2.1.8.1

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

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{3}{7} \cdot \frac{2}{3}}}{{(3^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 4.1.2.1.8.2

Cancel the common factor of $3$ .

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Step 4.1.2.1.8.2.1

Cancel the common factor.

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{3}{7} \cdot \frac{2}{3}}}{{(3^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 4.1.2.1.8.2.2

Rewrite the expression.

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{1}{7} \cdot 2}}{{(3^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 4.1.2.1.8.3

Combine $\frac{1}{7}$ and $2$ .

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}}}{{(3^{\frac{3}{7}})}^{\frac{2}{3}}}$

Step 4.1.2.1.9

Multiply the exponents in ${(3^{\frac{3}{7}})}^{\frac{2}{3}}$ .

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Step 4.1.2.1.9.1

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

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}}}{3^{\frac{3}{7} \cdot \frac{2}{3}}}$

Step 4.1.2.1.9.2

Cancel the common factor of $3$ .

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Step 4.1.2.1.9.2.1

Cancel the common factor.

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}}}{3^{\frac{3}{7} \cdot \frac{2}{3}}}$

Step 4.1.2.1.9.2.2

Rewrite the expression.

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}}}{3^{\frac{1}{7} \cdot 2}}$

Step 4.1.2.1.9.3

Combine $\frac{1}{7}$ and $2$ .

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}}}{3^{\frac{2}{7}}}$

Step 4.1.2.2

To write $- \frac{2^{\frac{2}{7}}}{3^{\frac{2}{7}}}$ as a fraction with a common denominator, multiply by $\frac{3^{\frac{14}{7}}}{3^{\frac{14}{7}}}$ .

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}}}{3^{\frac{2}{7}}} \cdot \frac{3^{\frac{14}{7}}}{3^{\frac{14}{7}}}$

Step 4.1.2.3

Write each expression with a common denominator of $3^{\frac{16}{7}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.3.1

Multiply $\frac{2^{\frac{2}{7}}}{3^{\frac{2}{7}}}$ by $\frac{3^{\frac{14}{7}}}{3^{\frac{14}{7}}}$ .

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}} \cdot 3^{\frac{14}{7}}}{3^{\frac{2}{7}} \cdot 3^{\frac{14}{7}}}$

Step 4.1.2.3.2

Multiply $3^{\frac{2}{7}}$ by $3^{\frac{14}{7}}$ by adding the exponents.

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Step 4.1.2.3.2.1

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

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}} \cdot 3^{\frac{14}{7}}}{3^{\frac{2}{7} + \frac{14}{7}}}$

Step 4.1.2.3.2.2

Combine the numerators over the common denominator.

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}} \cdot 3^{\frac{14}{7}}}{3^{\frac{2 + 14}{7}}}$

Step 4.1.2.3.2.3

Add $2$ and $14$ .

$\frac{2^{\frac{9}{7}}}{3^{\frac{16}{7}}} - \frac{2^{\frac{2}{7}} \cdot 3^{\frac{14}{7}}}{3^{\frac{16}{7}}}$

Step 4.1.2.4

Reduce the expression by cancelling the common factors.

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Step 4.1.2.4.1

Combine the numerators over the common denominator.

$\frac{2^{\frac{9}{7}} - 2^{\frac{2}{7}} \cdot 3^{\frac{14}{7}}}{3^{\frac{16}{7}}}$

Step 4.1.2.4.2

Cancel the common factor of $14$ and $7$ .

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Step 4.1.2.4.2.1

Factor $7$ out of $14$ .

$\frac{2^{\frac{9}{7}} - 2^{\frac{2}{7}} \cdot 3^{\frac{7 \cdot 2}{7}}}{3^{\frac{16}{7}}}$

Step 4.1.2.4.2.2

Cancel the common factors.

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Step 4.1.2.4.2.2.1

Factor $7$ out of $7$ .

$\frac{2^{\frac{9}{7}} - 2^{\frac{2}{7}} \cdot 3^{\frac{7 \cdot 2}{7 (1)}}}{3^{\frac{16}{7}}}$

Step 4.1.2.4.2.2.2

Cancel the common factor.

$\frac{2^{\frac{9}{7}} - 2^{\frac{2}{7}} \cdot 3^{\frac{7 \cdot 2}{7 \cdot 1}}}{3^{\frac{16}{7}}}$

Step 4.1.2.4.2.2.3

Rewrite the expression.

$\frac{2^{\frac{9}{7}} - 2^{\frac{2}{7}} \cdot 3^{\frac{2}{1}}}{3^{\frac{16}{7}}}$

Step 4.1.2.4.2.2.4

Divide $2$ by $1$ .

$\frac{2^{\frac{9}{7}} - 2^{\frac{2}{7}} \cdot 3^{2}}{3^{\frac{16}{7}}}$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Raise $3$ to the power of $2$ .

$\frac{2^{\frac{9}{7}} - 2^{\frac{2}{7}} \cdot 9}{3^{\frac{16}{7}}}$

Step 4.1.2.5.2

Multiply $9$ by $- 1$ .

$\frac{2^{\frac{9}{7}} - 9 \cdot 2^{\frac{2}{7}}}{3^{\frac{16}{7}}}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

$\frac{1}{3} \cdot {(0)}^{3} - {(0)}^{\frac{2}{3}}$

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

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

$\frac{1}{3} \cdot 0 - {(0)}^{\frac{2}{3}}$

Step 4.2.2.1.2

Multiply $\frac{1}{3}$ by $0$ .

$0 - {(0)}^{\frac{2}{3}}$

Step 4.2.2.1.3

Rewrite $0$ as $0^{3}$ .

$0 - {(0^{3})}^{\frac{2}{3}}$

Step 4.2.2.1.4

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

$0 - 0^{3 (\frac{2}{3})}$

Step 4.2.2.1.5

Cancel the common factor of $3$ .

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Step 4.2.2.1.5.1

Cancel the common factor.

$0 - 0^{3 (\frac{2}{3})}$

Step 4.2.2.1.5.2

Rewrite the expression.

$0 - 0^{2}$

Step 4.2.2.1.6

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

$0 - 0$

Step 4.2.2.1.7

Multiply $- 1$ by $0$ .

$0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$0$

Step 4.3

List all of the points.

$(\frac{2^{\frac{3}{7}}}{3^{\frac{3}{7}}}, \frac{2^{\frac{9}{7}} - 9 \cdot 2^{\frac{2}{7}}}{3^{\frac{16}{7}}}), (0, 0)$

Step 5