Calculus Examples

$y^{2} \frac{d y}{d x} = x^{- 3}$ , $y (2) = 0$

Step 1

Rewrite the equation.

$y^{2} d y = x^{- 3} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int y^{2} d y = \int x^{- 3} d x$

Step 2.2

By the Power Rule, the integral of $y^{2}$ with respect to $y$ is $\frac{1}{3} y^{3}$ .

$\frac{1}{3} y^{3} + C_{1} = \int x^{- 3} d x$

Step 2.3

Integrate the right side.

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Step 2.3.1

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{2} x^{- 2} + C_{2}$

Step 2.3.2

Simplify the answer.

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Step 2.3.2.1

Rewrite $- \frac{1}{2} x^{- 2} + C_{2}$ as $- \frac{1}{2} \cdot \frac{1}{x^{2}} + C_{2}$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{2} \cdot \frac{1}{x^{2}} + C_{2}$

Step 2.3.2.2

Simplify.

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Step 2.3.2.2.1

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

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{x^{2} \cdot 2} + C_{2}$

Step 2.3.2.2.2

Move $2$ to the left of $x^{2}$ .

$\frac{1}{3} y^{3} + C_{1} = - \frac{1}{2 x^{2}} + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\frac{1}{3} y^{3} = - \frac{1}{2 x^{2}} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $3$ .

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

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

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

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Step 3.2.1.1.1

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

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

Step 3.2.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.1.1.2.1

Cancel the common factor.

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

Step 3.2.1.1.2.2

Rewrite the expression.

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

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $3 (- \frac{1}{2 x^{2}} + K)$ .

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Step 3.2.2.1.1

Apply the distributive property.

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

Step 3.2.2.1.2

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

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Step 3.2.2.1.2.1

Multiply $- 1$ by $3$ .

$y^{3} = - 3 \frac{1}{2 x^{2}} + 3 K$

Step 3.2.2.1.2.2

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

$y^{3} = \frac{- 3}{2 x^{2}} + 3 K$

Step 3.2.2.1.3

Move the negative in front of the fraction.

$y^{3} = - \frac{3}{2 x^{2}} + 3 K$

Step 3.3

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

$y = \sqrt[3]{- \frac{3}{2 x^{2}} + 3 K}$

Step 3.4

Simplify $\sqrt[3]{- \frac{3}{2 x^{2}} + 3 K}$ .

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Step 3.4.1

Factor $3$ out of $- \frac{3}{2 x^{2}} + 3 K$ .

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Step 3.4.1.1

Factor $3$ out of $- \frac{3}{2 x^{2}}$ .

$y = \sqrt[3]{3 (- \frac{1}{2 x^{2}}) + 3 K}$

Step 3.4.1.2

Factor $3$ out of $3 K$ .

$y = \sqrt[3]{3 (- \frac{1}{2 x^{2}}) + 3 K}$

Step 3.4.1.3

Factor $3$ out of $3 (- \frac{1}{2 x^{2}}) + 3 K$ .

$y = \sqrt[3]{3 (- \frac{1}{2 x^{2}} + K)}$

Step 3.4.2

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

$y = \sqrt[3]{3 (- \frac{1}{2 x^{2}} + K \cdot \frac{2 x^{2}}{2 x^{2}})}$

Step 3.4.3

Simplify terms.

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Step 3.4.3.1

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

$y = \sqrt[3]{3 (- \frac{1}{2 x^{2}} + \frac{K (2 x^{2})}{2 x^{2}})}$

Step 3.4.3.2

Combine the numerators over the common denominator.

$y = \sqrt[3]{3 \frac{- 1 + K (2 x^{2})}{2 x^{2}}}$

Step 3.4.4

Move $2$ to the left of $K$ .

$y = \sqrt[3]{3 \frac{- 1 + 2 K x^{2}}{2 x^{2}}}$

Step 3.4.5

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

$y = \sqrt[3]{\frac{3 (- 1 + 2 K x^{2})}{2 x^{2}}}$

Step 3.4.6

Rewrite $\sqrt[3]{\frac{3 (- 1 + 2 K x^{2})}{2 x^{2}}}$ as $\frac{\sqrt[3]{3 (- 1 + 2 K x^{2})}}{\sqrt[3]{2 x^{2}}}$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})}}{\sqrt[3]{2 x^{2}}}$

Step 3.4.7

Multiply $\frac{\sqrt[3]{3 (- 1 + 2 K x^{2})}}{\sqrt[3]{2 x^{2}}}$ by $\frac{{\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})}}{\sqrt[3]{2 x^{2}}} \cdot \frac{{\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{2}}$

Step 3.4.8

Combine and simplify the denominator.

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Step 3.4.8.1

Multiply $\frac{\sqrt[3]{3 (- 1 + 2 K x^{2})}}{\sqrt[3]{2 x^{2}}}$ by $\frac{{\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{2}}$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{\sqrt[3]{2 x^{2}} {\sqrt[3]{2 x^{2}}}^{2}}$

Step 3.4.8.2

Raise $\sqrt[3]{2 x^{2}}$ to the power of $1$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{1} {\sqrt[3]{2 x^{2}}}^{2}}$

Step 3.4.8.3

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

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{1 + 2}}$

Step 3.4.8.4

Add $1$ and $2$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{{\sqrt[3]{2 x^{2}}}^{3}}$

Step 3.4.8.5

Rewrite ${\sqrt[3]{2 x^{2}}}^{3}$ as $2 x^{2}$ .

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Step 3.4.8.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{2 x^{2}}$ as ${(2 x^{2})}^{\frac{1}{3}}$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{{({(2 x^{2})}^{\frac{1}{3}})}^{3}}$

Step 3.4.8.5.2

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

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{{(2 x^{2})}^{\frac{1}{3} \cdot 3}}$

Step 3.4.8.5.3

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

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{{(2 x^{2})}^{\frac{3}{3}}}$

Step 3.4.8.5.4

Cancel the common factor of $3$ .

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Step 3.4.8.5.4.1

Cancel the common factor.

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{{(2 x^{2})}^{\frac{3}{3}}}$

Step 3.4.8.5.4.2

Rewrite the expression.

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{{(2 x^{2})}^{1}}$

Step 3.4.8.5.5

Simplify.

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} {\sqrt[3]{2 x^{2}}}^{2}}{2 x^{2}}$

Step 3.4.9

Simplify the numerator.

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Step 3.4.9.1

Rewrite ${\sqrt[3]{2 x^{2}}}^{2}$ as $\sqrt[3]{{(2 x^{2})}^{2}}$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} \sqrt[3]{{(2 x^{2})}^{2}}}{2 x^{2}}$

Step 3.4.9.2

Apply the product rule to $2 x^{2}$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} \sqrt[3]{2^{2} {(x^{2})}^{2}}}{2 x^{2}}$

Step 3.4.9.3

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

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} \sqrt[3]{4 {(x^{2})}^{2}}}{2 x^{2}}$

Step 3.4.9.4

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.4.9.4.1

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

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} \sqrt[3]{4 x^{2 \cdot 2}}}{2 x^{2}}$

Step 3.4.9.4.2

Multiply $2$ by $2$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} \sqrt[3]{4 x^{4}}}{2 x^{2}}$

Step 3.4.9.5

Rewrite $4 x^{4}$ as $x^{3} \cdot (4 x)$ .

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Step 3.4.9.5.1

Factor out $x^{3}$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} \sqrt[3]{4 (x^{3} x)}}{2 x^{2}}$

Step 3.4.9.5.2

Reorder $4$ and $x^{3}$ .

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} \sqrt[3]{x^{3} \cdot 4 x}}{2 x^{2}}$

Step 3.4.9.5.3

Add parentheses.

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} \sqrt[3]{x^{3} \cdot (4 x)}}{2 x^{2}}$

Step 3.4.9.6

Pull terms out from under the radical.

$y = \frac{\sqrt[3]{3 (- 1 + 2 K x^{2})} x \sqrt[3]{4 x}}{2 x^{2}}$

Step 3.4.9.7

Combine exponents.

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Step 3.4.9.7.1

Combine using the product rule for radicals.

$y = \frac{x \sqrt[3]{3 (- 1 + 2 K x^{2}) (4 x)}}{2 x^{2}}$

Step 3.4.9.7.2

Multiply $4$ by $3$ .

$y = \frac{x \sqrt[3]{12 (- 1 + 2 K x^{2}) x}}{2 x^{2}}$

Step 3.4.10

Reduce the expression by cancelling the common factors.

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Step 3.4.10.1

Cancel the common factor of $x$ and $x^{2}$ .

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Step 3.4.10.1.1

Factor $x$ out of $x \sqrt[3]{12 (- 1 + 2 K x^{2}) x}$ .

$y = \frac{x \sqrt[3]{12 (- 1 + 2 K x^{2}) x}}{2 x^{2}}$

Step 3.4.10.1.2

Cancel the common factors.

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Step 3.4.10.1.2.1

Factor $x$ out of $2 x^{2}$ .

$y = \frac{x \sqrt[3]{12 (- 1 + 2 K x^{2}) x}}{x (2 x)}$

Step 3.4.10.1.2.2

Cancel the common factor.

$y = \frac{x \sqrt[3]{12 (- 1 + 2 K x^{2}) x}}{x (2 x)}$

Step 3.4.10.1.2.3

Rewrite the expression.

$y = \frac{\sqrt[3]{12 (- 1 + 2 K x^{2}) x}}{2 x}$

Step 3.4.10.2

Reorder factors in $\frac{\sqrt[3]{12 (- 1 + 2 K x^{2}) x}}{2 x}$ .

$y = \frac{\sqrt[3]{12 x (- 1 + 2 K x^{2})}}{2 x}$

Step 4

Simplify the constant of integration.

$y = \frac{\sqrt[3]{12 x (- 1 + K x^{2})}}{2 x}$

Step 5

Use the initial condition to find the value of $K$ by substituting $2$ for $x$ and $0$ for $y$ in $y = \frac{\sqrt[3]{12 x (- 1 + K x^{2})}}{2 x}$ .

$0 = \frac{\sqrt[3]{12 \cdot 2 (- 1 + K \cdot 2^{2})}}{2 \cdot 2}$

Step 6

Solve for $K$ .

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Step 6.1

Set the numerator equal to zero.

$\sqrt[3]{12 \cdot (2 (- 1 + K \cdot 2^{2}))} = 0$

Step 6.2

Solve the equation for $K$ .

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Step 6.2.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{12 \cdot (2 (- 1 + K \cdot 2^{2}))}}^{3} = 0^{3}$

Step 6.2.2

Simplify each side of the equation.

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Step 6.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{12 \cdot (2 (- 1 + K \cdot 2^{2}))}$ as ${(12 \cdot (2 (- 1 + K \cdot 2^{2})))}^{\frac{1}{3}}$ .

${({(12 \cdot (2 (- 1 + K \cdot 2^{2})))}^{\frac{1}{3}})}^{3} = 0^{3}$

Step 6.2.2.2

Simplify the left side.

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Step 6.2.2.2.1

Simplify ${({(12 \cdot (2 (- 1 + K \cdot 2^{2})))}^{\frac{1}{3}})}^{3}$ .

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Step 6.2.2.2.1.1

Multiply the exponents in ${({(12 \cdot (2 (- 1 + K \cdot 2^{2})))}^{\frac{1}{3}})}^{3}$ .

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Step 6.2.2.2.1.1.1

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

${(12 \cdot (2 (- 1 + K \cdot 2^{2})))}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.2.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 6.2.2.2.1.1.2.1

Cancel the common factor.

${(12 \cdot (2 (- 1 + K \cdot 2^{2})))}^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 6.2.2.2.1.1.2.2

Rewrite the expression.

${(12 \cdot (2 (- 1 + K \cdot 2^{2})))}^{1} = 0^{3}$

Step 6.2.2.2.1.2

Simplify each term.

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Step 6.2.2.2.1.2.1

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

${(12 \cdot (2 (- 1 + K \cdot 4)))}^{1} = 0^{3}$

Step 6.2.2.2.1.2.2

Move $4$ to the left of $K$ .

${(12 \cdot (2 (- 1 + 4 K)))}^{1} = 0^{3}$

Step 6.2.2.2.1.3

Simplify by multiplying through.

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Step 6.2.2.2.1.3.1

Apply the distributive property.

${(12 \cdot (2 \cdot - 1 + 2 (4 K)))}^{1} = 0^{3}$

Step 6.2.2.2.1.3.2

Multiply.

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Step 6.2.2.2.1.3.2.1

Multiply $2$ by $- 1$ .

${(12 \cdot (- 2 + 2 (4 K)))}^{1} = 0^{3}$

Step 6.2.2.2.1.3.2.2

Multiply $4$ by $2$ .

${(12 \cdot (- 2 + 8 K))}^{1} = 0^{3}$

Step 6.2.2.2.1.3.3

Apply the distributive property.

${(12 \cdot - 2 + 12 (8 K))}^{1} = 0^{3}$

Step 6.2.2.2.1.3.4

Multiply.

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Step 6.2.2.2.1.3.4.1

Multiply $12$ by $- 2$ .

${(- 24 + 12 (8 K))}^{1} = 0^{3}$

Step 6.2.2.2.1.3.4.2

Multiply $8$ by $12$ .

${(- 24 + 96 K)}^{1} = 0^{3}$

Step 6.2.2.2.1.3.4.3

Simplify.

$- 24 + 96 K = 0^{3}$

Step 6.2.2.3

Simplify the right side.

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Step 6.2.2.3.1

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

$- 24 + 96 K = 0$

Step 6.2.3

Solve for $K$ .

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Step 6.2.3.1

Add $24$ to both sides of the equation.

$96 K = 24$

Step 6.2.3.2

Divide each term in $96 K = 24$ by $96$ and simplify.

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Step 6.2.3.2.1

Divide each term in $96 K = 24$ by $96$ .

$\frac{96 K}{96} = \frac{24}{96}$

Step 6.2.3.2.2

Simplify the left side.

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Step 6.2.3.2.2.1

Cancel the common factor of $96$ .

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Step 6.2.3.2.2.1.1

Cancel the common factor.

$\frac{96 K}{96} = \frac{24}{96}$

Step 6.2.3.2.2.1.2

Divide $K$ by $1$ .

$K = \frac{24}{96}$

Step 6.2.3.2.3

Simplify the right side.

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Step 6.2.3.2.3.1

Cancel the common factor of $24$ and $96$ .

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Step 6.2.3.2.3.1.1

Factor $24$ out of $24$ .

$K = \frac{24 (1)}{96}$

Step 6.2.3.2.3.1.2

Cancel the common factors.

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Step 6.2.3.2.3.1.2.1

Factor $24$ out of $96$ .

$K = \frac{24 \cdot 1}{24 \cdot 4}$

Step 6.2.3.2.3.1.2.2

Cancel the common factor.

$K = \frac{24 \cdot 1}{24 \cdot 4}$

Step 6.2.3.2.3.1.2.3

Rewrite the expression.

$K = \frac{1}{4}$

Step 7

Substitute $\frac{1}{4}$ for $K$ in $y = \frac{\sqrt[3]{12 x (- 1 + K x^{2})}}{2 x}$ and simplify.

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Step 7.1

Substitute $\frac{1}{4}$ for $K$ .

$y = \frac{\sqrt[3]{12 x (- 1 + \frac{1}{4} x^{2})}}{2 x}$

Step 7.2

Simplify the numerator.

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Step 7.2.1

Combine $\frac{1}{4}$ and $x^{2}$ .

$y = \frac{\sqrt[3]{12 x (- 1 + \frac{x^{2}}{4})}}{2 x}$

Step 7.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y = \frac{\sqrt[3]{12 x (- 1 \cdot \frac{4}{4} + \frac{x^{2}}{4})}}{2 x}$

Step 7.2.3

Combine $- 1$ and $\frac{4}{4}$ .

$y = \frac{\sqrt[3]{12 x (\frac{- 1 \cdot 4}{4} + \frac{x^{2}}{4})}}{2 x}$

Step 7.2.4

Combine the numerators over the common denominator.

$y = \frac{\sqrt[3]{12 x \frac{- 1 \cdot 4 + x^{2}}{4}}}{2 x}$

Step 7.2.5

Simplify the numerator.

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Step 7.2.5.1

Rewrite $4$ as $2^{2}$ .

$y = \frac{\sqrt[3]{12 x \frac{- 2^{2} + x^{2}}{4}}}{2 x}$

Step 7.2.5.2

Reorder $- 2^{2}$ and $x^{2}$ .

$y = \frac{\sqrt[3]{12 x \frac{x^{2} - 2^{2}}{4}}}{2 x}$

Step 7.2.5.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

$y = \frac{\sqrt[3]{12 x \frac{(x + 2) (x - 2)}{4}}}{2 x}$

Step 7.2.6

Combine exponents.

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Step 7.2.6.1

Combine $12$ and $\frac{(x + 2) (x - 2)}{4}$ .

$y = \frac{\sqrt[3]{x \frac{12 ((x + 2) (x - 2))}{4}}}{2 x}$

Step 7.2.6.2

Combine $x$ and $\frac{12 ((x + 2) (x - 2))}{4}$ .

$y = \frac{\sqrt[3]{\frac{x (12 ((x + 2) (x - 2)))}{4}}}{2 x}$

Step 7.2.7

Remove unnecessary parentheses.

$y = \frac{\sqrt[3]{\frac{x \cdot 12 (x + 2) (x - 2)}{4}}}{2 x}$

Step 7.2.8

Reduce the expression by cancelling the common factors.

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Step 7.2.8.1

Reduce the expression $\frac{x \cdot 12 (x + 2) (x - 2)}{4}$ by cancelling the common factors.

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Step 7.2.8.1.1

Factor $4$ out of $x \cdot 12 (x + 2) (x - 2)$ .

$y = \frac{\sqrt[3]{\frac{4 (x \cdot 3 (x + 2) (x - 2))}{4}}}{2 x}$

Step 7.2.8.1.2

Factor $4$ out of $4$ .

$y = \frac{\sqrt[3]{\frac{4 (x \cdot 3 (x + 2) (x - 2))}{4 (1)}}}{2 x}$

Step 7.2.8.1.3

Cancel the common factor.

$y = \frac{\sqrt[3]{\frac{4 (x \cdot 3 (x + 2) (x - 2))}{4 \cdot 1}}}{2 x}$

Step 7.2.8.1.4

Rewrite the expression.

$y = \frac{\sqrt[3]{\frac{x \cdot 3 (x + 2) (x - 2)}{1}}}{2 x}$

Step 7.2.8.2

Divide $x \cdot 3 (x + 2) (x - 2)$ by $1$ .

$y = \frac{\sqrt[3]{x \cdot 3 (x + 2) (x - 2)}}{2 x}$

Step 7.3

Reorder factors in $y = \frac{\sqrt[3]{x \cdot 3 (x + 2) (x - 2)}}{2 x}$ .

$y = \frac{\sqrt[3]{3 x (x + 2) (x - 2)}}{2 x}$