Calculus Examples

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

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

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = {(5 - x)}^{\frac{2}{3}}$ .

$x \frac{d}{d x} [{(5 - x)}^{\frac{2}{3}}] + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

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

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Step 1.1.2.1

To apply the Chain Rule, set $u$ as $5 - x$ .

$x (\frac{d}{d u} [u^{\frac{2}{3}}] \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

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 $n = \frac{2}{3}$ .

$x (\frac{2}{3} u^{\frac{2}{3} - 1} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.2.3

Replace all occurrences of $u$ with $5 - x$ .

$x (\frac{2}{3} {(5 - x)}^{\frac{2}{3} - 1} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.3

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

$x (\frac{2}{3} {(5 - x)}^{\frac{2}{3} - 1 \cdot \frac{3}{3}} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.4

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

$x (\frac{2}{3} {(5 - x)}^{\frac{2}{3} + \frac{- 1 \cdot 3}{3}} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.5

Combine the numerators over the common denominator.

$x (\frac{2}{3} {(5 - x)}^{\frac{2 - 1 \cdot 3}{3}} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Multiply $- 1$ by $3$ .

$x (\frac{2}{3} {(5 - x)}^{\frac{2 - 3}{3}} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.6.2

Subtract $3$ from $2$ .

$x (\frac{2}{3} {(5 - x)}^{\frac{- 1}{3}} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.7

Combine fractions.

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Step 1.1.7.1

Move the negative in front of the fraction.

$x (\frac{2}{3} {(5 - x)}^{- \frac{1}{3}} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.7.2

Combine $\frac{2}{3}$ and ${(5 - x)}^{- \frac{1}{3}}$ .

$x (\frac{2 {(5 - x)}^{- \frac{1}{3}}}{3} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.7.3

Move ${(5 - x)}^{- \frac{1}{3}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$x (\frac{2}{3 {(5 - x)}^{\frac{1}{3}}} \frac{d}{d x} [5 - x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.7.4

Combine $\frac{2}{3 {(5 - x)}^{\frac{1}{3}}}$ and $x$ .

$\frac{2 x}{3 {(5 - x)}^{\frac{1}{3}}} \frac{d}{d x} [5 - x] + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.8

By the Sum Rule, the derivative of $5 - x$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [- x]$ .

$\frac{2 x}{3 {(5 - x)}^{\frac{1}{3}}} (\frac{d}{d x} [5] + \frac{d}{d x} [- x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.9

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

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

Step 1.1.10

Add $0$ and $\frac{d}{d x} [- x]$ .

$\frac{2 x}{3 {(5 - x)}^{\frac{1}{3}}} \frac{d}{d x} [- x] + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.11

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

$\frac{2 x}{3 {(5 - x)}^{\frac{1}{3}}} (- \frac{d}{d x} [x]) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.12

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

$\frac{2 x}{3 {(5 - x)}^{\frac{1}{3}}} (- 1 \cdot 1) + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.13

Combine fractions.

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Step 1.1.13.1

Multiply $- 1$ by $1$ .

$\frac{2 x}{3 {(5 - x)}^{\frac{1}{3}}} \cdot - 1 + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.13.2

Combine $\frac{2 x}{3 {(5 - x)}^{\frac{1}{3}}}$ and $- 1$ .

$\frac{2 x \cdot - 1}{3 {(5 - x)}^{\frac{1}{3}}} + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.13.3

Simplify the expression.

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Step 1.1.13.3.1

Multiply $- 1$ by $2$ .

$\frac{- 2 x}{3 {(5 - x)}^{\frac{1}{3}}} + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.13.3.2

Move the negative in front of the fraction.

$- \frac{2 x}{3 {(5 - x)}^{\frac{1}{3}}} + {(5 - x)}^{\frac{2}{3}} \frac{d}{d x} [x]$

Step 1.1.14

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

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

Step 1.1.15

Multiply ${(5 - x)}^{\frac{2}{3}}$ by $1$ .

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

Step 1.1.16

To write ${(5 - x)}^{\frac{2}{3}}$ as a fraction with a common denominator, multiply by $\frac{3 {(5 - x)}^{\frac{1}{3}}}{3 {(5 - x)}^{\frac{1}{3}}}$ .

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

Step 1.1.17

Combine ${(5 - x)}^{\frac{2}{3}}$ and $\frac{3 {(5 - x)}^{\frac{1}{3}}}{3 {(5 - x)}^{\frac{1}{3}}}$ .

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

Step 1.1.18

Combine the numerators over the common denominator.

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

Step 1.1.19

Multiply ${(5 - x)}^{\frac{2}{3}}$ by ${(5 - x)}^{\frac{1}{3}}$ by adding the exponents.

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Step 1.1.19.1

Move ${(5 - x)}^{\frac{1}{3}}$ .

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

Step 1.1.19.2

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

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

Step 1.1.19.3

Combine the numerators over the common denominator.

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

Step 1.1.19.4

Add $1$ and $2$ .

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

Step 1.1.19.5

Divide $3$ by $3$ .

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

Step 1.1.20

Simplify ${(5 - x)}^{1} \cdot 3$ .

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

Step 1.1.21

Move $3$ to the left of $5 - x$ .

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

Step 1.1.22

Simplify.

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Step 1.1.22.1

Apply the distributive property.

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

Step 1.1.22.2

Simplify the numerator.

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Step 1.1.22.2.1

Simplify each term.

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Step 1.1.22.2.1.1

Multiply $3$ by $5$ .

$\frac{- 2 x + 15 + 3 (- x)}{3 {(5 - x)}^{\frac{1}{3}}}$

Step 1.1.22.2.1.2

Multiply $- 1$ by $3$ .

$\frac{- 2 x + 15 - 3 x}{3 {(5 - x)}^{\frac{1}{3}}}$

Step 1.1.22.2.2

Subtract $3 x$ from $- 2 x$ .

$\frac{- 5 x + 15}{3 {(5 - x)}^{\frac{1}{3}}}$

Step 1.1.22.3

Factor $5$ out of $- 5 x + 15$ .

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Step 1.1.22.3.1

Factor $5$ out of $- 5 x$ .

$\frac{5 (- x) + 15}{3 {(5 - x)}^{\frac{1}{3}}}$

Step 1.1.22.3.2

Factor $5$ out of $15$ .

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

Step 1.1.22.3.3

Factor $5$ out of $5 (- x) + 5 (3)$ .

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

Step 1.1.22.4

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

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

Step 1.1.22.5

Rewrite $3$ as $- 1 (- 3)$ .

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

Step 1.1.22.6

Factor $- 1$ out of $- (x) - 1 (- 3)$ .

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

Step 1.1.22.7

Rewrite $- (x - 3)$ as $- 1 (x - 3)$ .

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

Step 1.1.22.8

Move the negative in front of the fraction.

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

Step 1.2

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

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

Step 2

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

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Step 2.1

Set the first derivative equal to $0$ .

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

Step 2.2

Set the numerator equal to zero.

$5 (x - 3) = 0$

Step 2.3

Solve the equation for $x$ .

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Step 2.3.1

Divide each term in $5 (x - 3) = 0$ by $5$ and simplify.

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Step 2.3.1.1

Divide each term in $5 (x - 3) = 0$ by $5$ .

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

Step 2.3.1.2

Simplify the left side.

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Step 2.3.1.2.1

Cancel the common factor of $5$ .

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Step 2.3.1.2.1.1

Cancel the common factor.

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

Step 2.3.1.2.1.2

Divide $x - 3$ by $1$ .

$x - 3 = \frac{0}{5}$

Step 2.3.1.3

Simplify the right side.

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Step 2.3.1.3.1

Divide $0$ by $5$ .

$x - 3 = 0$

Step 2.3.2

Add $3$ to both sides of the equation.

$x = 3$

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.

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

Step 3.1.2

Anything raised to $1$ is the base itself.

$- \frac{5 (x - 3)}{3 \sqrt[3]{5 - x}}$

Step 3.2

Set the denominator in $\frac{5 (x - 3)}{3 \sqrt[3]{5 - x}}$ equal to $0$ to find where the expression is undefined.

$3 \sqrt[3]{5 - x} = 0$

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.

${(3 \sqrt[3]{5 - x})}^{3} = 0^{3}$

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 $\sqrt[3]{5 - x}$ as ${(5 - x)}^{\frac{1}{3}}$ .

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

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Simplify ${(3 {(5 - x)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.1

Apply the product rule to $3 {(5 - x)}^{\frac{1}{3}}$ .

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

Step 3.3.2.2.1.2

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

$27 {({(5 - x)}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 3.3.2.2.1.3

Multiply the exponents in ${({(5 - x)}^{\frac{1}{3}})}^{3}$ .

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Step 3.3.2.2.1.3.1

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

$27 {(5 - x)}^{\frac{1}{3} \cdot 3} = 0^{3}$

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.

$27 {(5 - x)}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 3.3.2.2.1.3.2.2

Rewrite the expression.

$27 {(5 - x)}^{1} = 0^{3}$

Step 3.3.2.2.1.4

Simplify.

$27 (5 - x) = 0^{3}$

Step 3.3.2.2.1.5

Apply the distributive property.

$27 \cdot 5 + 27 (- x) = 0^{3}$

Step 3.3.2.2.1.6

Multiply.

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Step 3.3.2.2.1.6.1

Multiply $27$ by $5$ .

$135 + 27 (- x) = 0^{3}$

Step 3.3.2.2.1.6.2

Multiply $- 1$ by $27$ .

$135 - 27 x = 0^{3}$

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 $0$ .

$135 - 27 x = 0$

Step 3.3.3

Solve for $x$ .

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Step 3.3.3.1

Subtract $135$ from both sides of the equation.

$- 27 x = - 135$

Step 3.3.3.2

Divide each term in $- 27 x = - 135$ by $- 27$ and simplify.

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Step 3.3.3.2.1

Divide each term in $- 27 x = - 135$ by $- 27$ .

$\frac{- 27 x}{- 27} = \frac{- 135}{- 27}$

Step 3.3.3.2.2

Simplify the left side.

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Step 3.3.3.2.2.1

Cancel the common factor of $- 27$ .

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Step 3.3.3.2.2.1.1

Cancel the common factor.

$\frac{- 27 x}{- 27} = \frac{- 135}{- 27}$

Step 3.3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 135}{- 27}$

Step 3.3.3.2.3

Simplify the right side.

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Step 3.3.3.2.3.1

Divide $- 135$ by $- 27$ .

$x = 5$

Step 4

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

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Step 4.1

Evaluate at $x = 3$ .

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Step 4.1.1

Substitute $3$ for $x$ .

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

Step 4.1.2

Simplify.

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Step 4.1.2.1

Multiply $- 1$ by $3$ .

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

Step 4.1.2.2

Subtract $3$ from $5$ .

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

Step 4.2

Evaluate at $x = 5$ .

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Step 4.2.1

Substitute $5$ for $x$ .

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

Step 4.2.2

Simplify.

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Step 4.2.2.1

Simplify the expression.

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Step 4.2.2.1.1

Multiply $- 1$ by $5$ .

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

Step 4.2.2.1.2

Subtract $5$ from $5$ .

$5 \cdot 0^{\frac{2}{3}}$

Step 4.2.2.1.3

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

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

Step 4.2.2.1.4

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

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

Step 4.2.2.2

Cancel the common factor of $3$ .

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Step 4.2.2.2.1

Cancel the common factor.

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

Step 4.2.2.2.2

Rewrite the expression.

$5 \cdot 0^{2}$

Step 4.2.2.3

Simplify the expression.

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Step 4.2.2.3.1

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

$5 \cdot 0$

Step 4.2.2.3.2

Multiply $5$ by $0$ .

$0$

Step 4.3

List all of the points.

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

Step 5