Calculus Examples

$f (x) = x^{\frac{2}{3}} - x^{\frac{5}{3}}$

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 $x^{\frac{2}{3}} - x^{\frac{5}{3}}$ with respect to $x$ is $\frac{d}{d x} [x^{\frac{2}{3}}] + \frac{d}{d x} [- x^{\frac{5}{3}}]$ .

$f' (x) = \frac{d}{d x} (x^{\frac{2}{3}}) + \frac{d}{d x} (- x^{\frac{5}{3}})$

Step 1.1.2

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

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Step 1.1.2.1

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

$f' (x) = \frac{2}{3} x^{\frac{2}{3} - 1} + \frac{d}{d x} (- x^{\frac{5}{3}})$

Step 1.1.2.2

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

$f' (x) = \frac{2}{3} x^{\frac{2}{3} - 1 \cdot \frac{3}{3}} + \frac{d}{d x} (- x^{\frac{5}{3}})$

Step 1.1.2.3

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

$f' (x) = \frac{2}{3} x^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} + \frac{d}{d x} (- x^{\frac{5}{3}})$

Step 1.1.2.4

Combine the numerators over the common denominator.

$f' (x) = \frac{2}{3} x^{\frac{2 - 1 \cdot 3}{3}} + \frac{d}{d x} (- x^{\frac{5}{3}})$

Step 1.1.2.5

Simplify the numerator.

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Step 1.1.2.5.1

Multiply $- 1$ by $3$ .

$f' (x) = \frac{2}{3} x^{\frac{2 - 3}{3}} + \frac{d}{d x} (- x^{\frac{5}{3}})$

Step 1.1.2.5.2

Subtract $3$ from $2$ .

$f' (x) = \frac{2}{3} x^{\frac{- 1}{3}} + \frac{d}{d x} (- x^{\frac{5}{3}})$

Step 1.1.2.6

Move the negative in front of the fraction.

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} + \frac{d}{d x} (- x^{\frac{5}{3}})$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x^{\frac{5}{3}}]$ .

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Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{\frac{5}{3}}$ with respect to $x$ is $- \frac{d}{d x} [x^{\frac{5}{3}}]$ .

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} - \frac{d}{d x} x^{\frac{5}{3}}$

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 $n = \frac{5}{3}$ .

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} - (\frac{5}{3} x^{\frac{5}{3} - 1})$

Step 1.1.3.3

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

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} - (\frac{5}{3} x^{\frac{5}{3} - 1 \cdot \frac{3}{3}})$

Step 1.1.3.4

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

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} - (\frac{5}{3} x^{\frac{5}{3} + \frac{- 1 \cdot 3}{3}})$

Step 1.1.3.5

Combine the numerators over the common denominator.

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} - (\frac{5}{3} x^{\frac{5 - 1 \cdot 3}{3}})$

Step 1.1.3.6

Simplify the numerator.

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Step 1.1.3.6.1

Multiply $- 1$ by $3$ .

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} - (\frac{5}{3} x^{\frac{5 - 3}{3}})$

Step 1.1.3.6.2

Subtract $3$ from $5$ .

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} - (\frac{5}{3} x^{\frac{2}{3}})$

Step 1.1.3.7

Combine $\frac{5}{3}$ and $x^{\frac{2}{3}}$ .

$f' (x) = \frac{2}{3} x^{- \frac{1}{3}} - \frac{5 x^{\frac{2}{3}}}{3}$

Step 1.1.4

Simplify.

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Step 1.1.4.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$f' (x) = \frac{2}{3} \cdot \frac{1}{x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3}$

Step 1.1.4.2

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

$f' (x) = \frac{2}{3 x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3}$

Step 1.2

The first derivative of $f (x)$ with respect to $x$ is $\frac{2}{3 x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3}$ .

$\frac{2}{3 x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3}$

Step 2

Set the first derivative equal to $0$ then solve the equation $\frac{2}{3 x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3} = 0$ .

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Step 2.1

Set the first derivative equal to $0$ .

$\frac{2}{3 x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3} = 0$

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.

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

Step 2.2.2

Since $3 x^{\frac{1}{3}}, 3, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $3, 3, 1$ then find LCM for the variable part $x^{\frac{1}{3}}$ .

Step 2.2.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

Step 2.2.4

Since $3$ has no factors besides $1$ and $3$ .

$3$ is a prime number

Step 2.2.5

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

Step 2.2.6

The LCM of $3, 3, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$3$

Step 2.2.7

The LCM of $x^{\frac{1}{3}}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

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

Step 2.2.8

The LCM for $3 x^{\frac{1}{3}}, 3, 1$ is the numeric part $3$ multiplied by the variable part.

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

Step 2.3

Multiply each term in $\frac{2}{3 x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3} = 0$ by $3 x^{\frac{1}{3}}$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{2}{3 x^{\frac{1}{3}}} - \frac{5 x^{\frac{2}{3}}}{3} = 0$ by $3 x^{\frac{1}{3}}$ .

$\frac{2}{3 x^{\frac{1}{3}}} (3 x^{\frac{1}{3}}) - \frac{5 x^{\frac{2}{3}}}{3} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

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.

$3 \frac{2}{3 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{5 x^{\frac{2}{3}}}{3} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2

Cancel the common factor of $3$ .

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Step 2.3.2.1.2.1

Cancel the common factor.

$3 \frac{2}{3 x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{5 x^{\frac{2}{3}}}{3} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.2.2

Rewrite the expression.

$\frac{2}{x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{5 x^{\frac{2}{3}}}{3} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.3

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

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Step 2.3.2.1.3.1

Cancel the common factor.

$\frac{2}{x^{\frac{1}{3}}} x^{\frac{1}{3}} - \frac{5 x^{\frac{2}{3}}}{3} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.3.2

Rewrite the expression.

$2 - \frac{5 x^{\frac{2}{3}}}{3} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.4

Cancel the common factor of $3$ .

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Step 2.3.2.1.4.1

Move the leading negative in $- \frac{5 x^{\frac{2}{3}}}{3}$ into the numerator.

$2 + \frac{- 5 x^{\frac{2}{3}}}{3} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.4.2

Factor $3$ out of $3 x^{\frac{1}{3}}$ .

$2 + \frac{- 5 x^{\frac{2}{3}}}{3} (3 (x^{\frac{1}{3}})) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.4.3

Cancel the common factor.

$2 + \frac{- 5 x^{\frac{2}{3}}}{3} (3 x^{\frac{1}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.4.4

Rewrite the expression.

$2 - 5 x^{\frac{2}{3}} x^{\frac{1}{3}} = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.5

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

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Step 2.3.2.1.5.1

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

$2 - 5 (x^{\frac{1}{3}} x^{\frac{2}{3}}) = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.5.2

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

$2 - 5 x^{\frac{1}{3} + \frac{2}{3}} = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.5.3

Combine the numerators over the common denominator.

$2 - 5 x^{\frac{1 + 2}{3}} = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.5.4

Add $1$ and $2$ .

$2 - 5 x^{\frac{3}{3}} = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.5.5

Divide $3$ by $3$ .

$2 - 5 x^{1} = 0 (3 x^{\frac{1}{3}})$

Step 2.3.2.1.6

Simplify $- 5 x^{1}$ .

$2 - 5 x = 0 (3 x^{\frac{1}{3}})$

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 $0$ .

$2 - 5 x = 0 x^{\frac{1}{3}}$

Step 2.3.3.1.2

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

$2 - 5 x = 0$

Step 2.4

Solve the equation.

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Step 2.4.1

Subtract $2$ from both sides of the equation.

$- 5 x = - 2$

Step 2.4.2

Divide each term in $- 5 x = - 2$ by $- 5$ and simplify.

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Step 2.4.2.1

Divide each term in $- 5 x = - 2$ by $- 5$ .

$\frac{- 5 x}{- 5} = \frac{- 2}{- 5}$

Step 2.4.2.2

Simplify the left side.

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Step 2.4.2.2.1

Cancel the common factor of $- 5$ .

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Step 2.4.2.2.1.1

Cancel the common factor.

$\frac{- 5 x}{- 5} = \frac{- 2}{- 5}$

Step 2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 5}$

Step 2.4.2.3

Simplify the right side.

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Step 2.4.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{2}{5}$

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.

$\frac{2}{3 \sqrt[3]{x^{1}}} - \frac{5 x^{\frac{2}{3}}}{3}$

Step 3.1.2

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

$\frac{2}{3 \sqrt[3]{x^{1}}} - \frac{5 \sqrt[3]{x^{2}}}{3}$

Step 3.1.3

Anything raised to $1$ is the base itself.

$\frac{2}{3 \sqrt[3]{x}} - \frac{5 \sqrt[3]{x^{2}}}{3}$

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 $x^{\frac{2}{3}} - x^{\frac{5}{3}}$ at each $x$ value where the derivative is $0$ or undefined.

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Step 4.1

Evaluate at $x = \frac{2}{5}$ .

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Step 4.1.1

Substitute $\frac{2}{5}$ for $x$ .

${(\frac{2}{5})}^{\frac{2}{3}} - {(\frac{2}{5})}^{\frac{5}{3}}$

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 $\frac{2}{5}$ .

$\frac{2^{\frac{2}{3}}}{5^{\frac{2}{3}}} - {(\frac{2}{5})}^{\frac{5}{3}}$

Step 4.1.2.1.2

Apply the product rule to $\frac{2}{5}$ .

$\frac{2^{\frac{2}{3}}}{5^{\frac{2}{3}}} - \frac{2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.2

To write $\frac{2^{\frac{2}{3}}}{5^{\frac{2}{3}}}$ as a fraction with a common denominator, multiply by $\frac{5^{\frac{3}{3}}}{5^{\frac{3}{3}}}$ .

$\frac{2^{\frac{2}{3}}}{5^{\frac{2}{3}}} \cdot \frac{5^{\frac{3}{3}}}{5^{\frac{3}{3}}} - \frac{2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.3

Write each expression with a common denominator of $5^{\frac{5}{3}}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1.2.3.1

Multiply $\frac{2^{\frac{2}{3}}}{5^{\frac{2}{3}}}$ by $\frac{5^{\frac{3}{3}}}{5^{\frac{3}{3}}}$ .

$\frac{2^{\frac{2}{3}} \cdot 5^{\frac{3}{3}}}{5^{\frac{2}{3}} \cdot 5^{\frac{3}{3}}} - \frac{2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.3.2

Multiply $5^{\frac{2}{3}}$ by $5^{\frac{3}{3}}$ 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.

$\frac{2^{\frac{2}{3}} \cdot 5^{\frac{3}{3}}}{5^{\frac{2}{3} + \frac{3}{3}}} - \frac{2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.3.2.2

Combine the numerators over the common denominator.

$\frac{2^{\frac{2}{3}} \cdot 5^{\frac{3}{3}}}{5^{\frac{2 + 3}{3}}} - \frac{2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.3.2.3

Add $2$ and $3$ .

$\frac{2^{\frac{2}{3}} \cdot 5^{\frac{3}{3}}}{5^{\frac{5}{3}}} - \frac{2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

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.

$\frac{2^{\frac{2}{3}} \cdot 5^{\frac{3}{3}} - 2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.4.2

Cancel the common factor of $3$ .

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Step 4.1.2.4.2.1

Cancel the common factor.

$\frac{2^{\frac{2}{3}} \cdot 5^{\frac{3}{3}} - 2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.4.2.2

Rewrite the expression.

$\frac{2^{\frac{2}{3}} \cdot 5^{1} - 2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.5

Simplify the numerator.

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Step 4.1.2.5.1

Evaluate the exponent.

$\frac{2^{\frac{2}{3}} \cdot 5 - 2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.1.2.5.2

Move $5$ to the left of $2^{\frac{2}{3}}$ .

$\frac{5 \cdot 2^{\frac{2}{3}} - 2^{\frac{5}{3}}}{5^{\frac{5}{3}}}$

Step 4.2

Evaluate at $x = 0$ .

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Step 4.2.1

Substitute $0$ for $x$ .

${(0)}^{\frac{2}{3}} - {(0)}^{\frac{5}{3}}$

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

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

${(0^{3})}^{\frac{2}{3}} - {(0)}^{\frac{5}{3}}$

Step 4.2.2.1.2

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

$0^{3 (\frac{2}{3})} - {(0)}^{\frac{5}{3}}$

Step 4.2.2.1.3

Cancel the common factor of $3$ .

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Step 4.2.2.1.3.1

Cancel the common factor.

$0^{3 (\frac{2}{3})} - {(0)}^{\frac{5}{3}}$

Step 4.2.2.1.3.2

Rewrite the expression.

$0^{2} - {(0)}^{\frac{5}{3}}$

Step 4.2.2.1.4

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

$0 - {(0)}^{\frac{5}{3}}$

Step 4.2.2.1.5

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

$0 - {(0^{3})}^{\frac{5}{3}}$

Step 4.2.2.1.6

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

$0 - 0^{3 (\frac{5}{3})}$

Step 4.2.2.1.7

Cancel the common factor of $3$ .

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Step 4.2.2.1.7.1

Cancel the common factor.

$0 - 0^{3 (\frac{5}{3})}$

Step 4.2.2.1.7.2

Rewrite the expression.

$0 - 0^{5}$

Step 4.2.2.1.8

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

$0 - 0$

Step 4.2.2.1.9

Multiply $- 1$ by $0$ .

$0 + 0$

Step 4.2.2.2

Add $0$ and $0$ .

$0$

Step 4.3

List all of the points.

$(\frac{2}{5}, \frac{5 \cdot 2^{\frac{2}{3}} - 2^{\frac{5}{3}}}{5^{\frac{5}{3}}}), (0, 0)$

Step 5