Calculus Examples

$2 \sqrt{y} \frac{d y}{d x} - 3 x = 0$

Step 1

Separate the variables.

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Step 1.1

Solve for $\frac{d y}{d x}$ .

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Step 1.1.1

Add $3 x$ to both sides of the equation.

$2 \sqrt{y} \frac{d y}{d x} = 3 x$

Step 1.1.2

Divide each term in $2 \sqrt{y} \frac{d y}{d x} = 3 x$ by $2 \sqrt{y}$ and simplify.

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Step 1.1.2.1

Divide each term in $2 \sqrt{y} \frac{d y}{d x} = 3 x$ by $2 \sqrt{y}$ .

$\frac{2 \sqrt{y} \frac{d y}{d x}}{2 \sqrt{y}} = \frac{3 x}{2 \sqrt{y}}$

Step 1.1.2.2

Simplify the left side.

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Step 1.1.2.2.1

Cancel the common factor of $2$ .

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Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{2 \sqrt{y} \frac{d y}{d x}}{2 \sqrt{y}} = \frac{3 x}{2 \sqrt{y}}$

Step 1.1.2.2.1.2

Rewrite the expression.

$\frac{\sqrt{y} \frac{d y}{d x}}{\sqrt{y}} = \frac{3 x}{2 \sqrt{y}}$

Step 1.1.2.2.2

Cancel the common factor of $\sqrt{y}$ .

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Step 1.1.2.2.2.1

Cancel the common factor.

$\frac{\sqrt{y} \frac{d y}{d x}}{\sqrt{y}} = \frac{3 x}{2 \sqrt{y}}$

Step 1.1.2.2.2.2

Divide $\frac{d y}{d x}$ by $1$ .

$\frac{d y}{d x} = \frac{3 x}{2 \sqrt{y}}$

Step 1.1.2.3

Simplify the right side.

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Step 1.1.2.3.1

Multiply $\frac{3 x}{2 \sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\frac{d y}{d x} = \frac{3 x}{2 \sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}}$

Step 1.1.2.3.2

Combine and simplify the denominator.

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Step 1.1.2.3.2.1

Multiply $\frac{3 x}{2 \sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 \sqrt{y} \sqrt{y}}$

Step 1.1.2.3.2.2

Move $\sqrt{y}$ .

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 (\sqrt{y} \sqrt{y})}$

Step 1.1.2.3.2.3

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 ({\sqrt{y}}^{1} \sqrt{y})}$

Step 1.1.2.3.2.4

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 ({\sqrt{y}}^{1} {\sqrt{y}}^{1})}$

Step 1.1.2.3.2.5

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

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 {\sqrt{y}}^{1 + 1}}$

Step 1.1.2.3.2.6

Add $1$ and $1$ .

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 {\sqrt{y}}^{2}}$

Step 1.1.2.3.2.7

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 1.1.2.3.2.7.1

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

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 {(y^{\frac{1}{2}})}^{2}}$

Step 1.1.2.3.2.7.2

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

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 y^{\frac{1}{2} \cdot 2}}$

Step 1.1.2.3.2.7.3

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

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 y^{\frac{2}{2}}}$

Step 1.1.2.3.2.7.4

Cancel the common factor of $2$ .

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Step 1.1.2.3.2.7.4.1

Cancel the common factor.

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 y^{\frac{2}{2}}}$

Step 1.1.2.3.2.7.4.2

Rewrite the expression.

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 y^{1}}$

Step 1.1.2.3.2.7.5

Simplify.

$\frac{d y}{d x} = \frac{3 x \sqrt{y}}{2 y}$

Step 1.2

Regroup factors.

$\frac{d y}{d x} = \frac{3 x}{2} \cdot \frac{\sqrt{y}}{y}$

Step 1.3

Multiply both sides by $\frac{y}{\sqrt{y}}$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y}{\sqrt{y}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4

Simplify.

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Step 1.4.1

Multiply $\frac{y}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y}{\sqrt{y}} \cdot \frac{\sqrt{y}}{\sqrt{y}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2

Combine and simplify the denominator.

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Step 1.4.2.1

Multiply $\frac{y}{\sqrt{y}}$ by $\frac{\sqrt{y}}{\sqrt{y}}$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{\sqrt{y} \sqrt{y}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.2

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{{\sqrt{y}}^{1} \sqrt{y}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.3

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{{\sqrt{y}}^{1} {\sqrt{y}}^{1}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.4

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

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{{\sqrt{y}}^{1 + 1}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.5

Add $1$ and $1$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{{\sqrt{y}}^{2}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 1.4.2.6.1

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

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{{(y^{\frac{1}{2}})}^{2}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.6.2

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

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{y^{\frac{1}{2} \cdot 2}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.6.3

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

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{y^{\frac{2}{2}}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.6.4

Cancel the common factor of $2$ .

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Step 1.4.2.6.4.1

Cancel the common factor.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{y^{\frac{2}{2}}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.6.4.2

Rewrite the expression.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{y^{1}} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.2.6.5

Simplify.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{y} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.3

Cancel the common factor of $y$ .

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Step 1.4.3.1

Cancel the common factor.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{y \sqrt{y}}{y} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.3.2

Divide $\sqrt{y}$ by $1$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \sqrt{y} (\frac{3 x}{2} \cdot \frac{\sqrt{y}}{y})$

Step 1.4.4

Multiply $\frac{3 x}{2}$ by $\frac{\sqrt{y}}{y}$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \sqrt{y} \frac{3 x \sqrt{y}}{2 y}$

Step 1.4.5

Multiply $\sqrt{y} \frac{3 x \sqrt{y}}{2 y}$ .

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Step 1.4.5.1

Combine $\sqrt{y}$ and $\frac{3 x \sqrt{y}}{2 y}$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{\sqrt{y} (3 x \sqrt{y})}{2 y}$

Step 1.4.5.2

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x ({\sqrt{y}}^{1} \sqrt{y})}{2 y}$

Step 1.4.5.3

Raise $\sqrt{y}$ to the power of $1$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x ({\sqrt{y}}^{1} {\sqrt{y}}^{1})}{2 y}$

Step 1.4.5.4

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

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x {\sqrt{y}}^{1 + 1}}{2 y}$

Step 1.4.5.5

Add $1$ and $1$ .

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x {\sqrt{y}}^{2}}{2 y}$

Step 1.4.6

Rewrite ${\sqrt{y}}^{2}$ as $y$ .

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Step 1.4.6.1

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

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x {(y^{\frac{1}{2}})}^{2}}{2 y}$

Step 1.4.6.2

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

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x y^{\frac{1}{2} \cdot 2}}{2 y}$

Step 1.4.6.3

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

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x y^{\frac{2}{2}}}{2 y}$

Step 1.4.6.4

Cancel the common factor of $2$ .

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Step 1.4.6.4.1

Cancel the common factor.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x y^{\frac{2}{2}}}{2 y}$

Step 1.4.6.4.2

Rewrite the expression.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x y^{1}}{2 y}$

Step 1.4.6.5

Simplify.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x y}{2 y}$

Step 1.4.7

Cancel the common factor of $y$ .

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Step 1.4.7.1

Cancel the common factor.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x y}{2 y}$

Step 1.4.7.2

Rewrite the expression.

$\frac{y}{\sqrt{y}} \frac{d y}{d x} = \frac{3 x}{2}$

Step 1.5

Rewrite the equation.

$\frac{y}{\sqrt{y}} d y = \frac{3 x}{2} d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \frac{y}{\sqrt{y}} d y = \int \frac{3 x}{2} d x$

Step 2.2

Integrate the left side.

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Step 2.2.1

Simplify the expression.

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Step 2.2.1.1

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

$\int \frac{y}{y^{\frac{1}{2}}} d y = \int \frac{3 x}{2} d x$

Step 2.2.1.2

Simplify.

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Step 2.2.1.2.1

Move $y^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\int y \cdot y^{- \frac{1}{2}} d y = \int \frac{3 x}{2} d x$

Step 2.2.1.2.2

Multiply $y$ by $y^{- \frac{1}{2}}$ by adding the exponents.

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Step 2.2.1.2.2.1

Multiply $y$ by $y^{- \frac{1}{2}}$ .

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Step 2.2.1.2.2.1.1

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

$\int y^{1} y^{- \frac{1}{2}} d y = \int \frac{3 x}{2} d x$

Step 2.2.1.2.2.1.2

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

$\int y^{1 - \frac{1}{2}} d y = \int \frac{3 x}{2} d x$

Step 2.2.1.2.2.2

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

$\int y^{\frac{2}{2} - \frac{1}{2}} d y = \int \frac{3 x}{2} d x$

Step 2.2.1.2.2.3

Combine the numerators over the common denominator.

$\int y^{\frac{2 - 1}{2}} d y = \int \frac{3 x}{2} d x$

Step 2.2.1.2.2.4

Subtract $1$ from $2$ .

$\int y^{\frac{1}{2}} d y = \int \frac{3 x}{2} d x$

Step 2.2.2

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

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

Step 2.3

Integrate the right side.

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Step 2.3.1

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

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

Step 2.3.2

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

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

Step 2.3.3

Simplify the answer.

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Step 2.3.3.1

Rewrite $\frac{3}{2} (\frac{1}{2} x^{2} + C_{2})$ as $\frac{3}{2} \cdot \frac{1}{2} x^{2} + C_{2}$ .

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

Step 2.3.3.2

Simplify.

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Step 2.3.3.2.1

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

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

Step 2.3.3.2.2

Multiply $2$ by $2$ .

$\frac{2}{3} y^{\frac{3}{2}} + C_{1} = \frac{3}{4} x^{2} + C_{2}$

Step 2.4

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

$\frac{2}{3} y^{\frac{3}{2}} = \frac{3}{4} x^{2} + K$

Step 3

Solve for $y$ .

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Step 3.1

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} (\frac{2}{3} y^{\frac{3}{2}}) = \frac{3}{2} (\frac{3}{4} x^{2} + K)$

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 $\frac{3}{2} (\frac{2}{3} y^{\frac{3}{2}})$ .

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Step 3.2.1.1.1

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

$\frac{3}{2} \cdot \frac{2 y^{\frac{3}{2}}}{3} = \frac{3}{2} (\frac{3}{4} x^{2} + K)$

Step 3.2.1.1.2

Combine.

$\frac{3 (2 y^{\frac{3}{2}})}{2 \cdot 3} = \frac{3}{2} (\frac{3}{4} x^{2} + K)$

Step 3.2.1.1.3

Cancel the common factor.

$\frac{3 (2 y^{\frac{3}{2}})}{2 \cdot 3} = \frac{3}{2} (\frac{3}{4} x^{2} + K)$

Step 3.2.1.1.4

Rewrite the expression.

$\frac{2 y^{\frac{3}{2}}}{2} = \frac{3}{2} (\frac{3}{4} x^{2} + K)$

Step 3.2.1.1.5

Cancel the common factor.

$\frac{2 y^{\frac{3}{2}}}{2} = \frac{3}{2} (\frac{3}{4} x^{2} + K)$

Step 3.2.1.1.6

Divide $y^{\frac{3}{2}}$ by $1$ .

$y^{\frac{3}{2}} = \frac{3}{2} (\frac{3}{4} x^{2} + K)$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{3}{2} (\frac{3}{4} x^{2} + K)$ .

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Step 3.2.2.1.1

Simplify terms.

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Step 3.2.2.1.1.1

Combine $\frac{3}{4}$ and $x^{2}$ .

$y^{\frac{3}{2}} = \frac{3}{2} (\frac{3 x^{2}}{4} + K)$

Step 3.2.2.1.1.2

Apply the distributive property.

$y^{\frac{3}{2}} = \frac{3}{2} \cdot \frac{3 x^{2}}{4} + \frac{3}{2} K$

Step 3.2.2.1.1.3

Combine.

$y^{\frac{3}{2}} = \frac{3 (3 x^{2})}{2 \cdot 4} + \frac{3}{2} K$

Step 3.2.2.1.1.4

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

$y^{\frac{3}{2}} = \frac{3 (3 x^{2})}{2 \cdot 4} + \frac{3 K}{2}$

Step 3.2.2.1.2

Simplify each term.

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Step 3.2.2.1.2.1

Multiply $3$ by $3$ .

$y^{\frac{3}{2}} = \frac{3^{2} x^{2}}{2 \cdot 4} + \frac{3 K}{2}$

Step 3.2.2.1.2.2

Multiply $2$ by $4$ .

$y^{\frac{3}{2}} = \frac{3^{2} x^{2}}{8} + \frac{3 K}{2}$

Step 3.2.2.1.2.3

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

$y^{\frac{3}{2}} = \frac{9 x^{2}}{8} + \frac{3 K}{2}$

Step 3.3

Raise each side of the equation to the power of $\frac{2}{3}$ to eliminate the fractional exponent on the left side.

${(y^{\frac{3}{2}})}^{\frac{2}{3}} = {(\frac{9 x^{2}}{8} + \frac{3 K}{2})}^{\frac{2}{3}}$

Step 3.4

Simplify the left side.

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Step 3.4.1

Simplify ${(y^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 3.4.1.1

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

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Step 3.4.1.1.1

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

$y^{\frac{3}{2} \cdot \frac{2}{3}} = {(\frac{9 x^{2}}{8} + \frac{3 K}{2})}^{\frac{2}{3}}$

Step 3.4.1.1.2

Cancel the common factor of $3$ .

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Step 3.4.1.1.2.1

Cancel the common factor.

$y^{\frac{3}{2} \cdot \frac{2}{3}} = {(\frac{9 x^{2}}{8} + \frac{3 K}{2})}^{\frac{2}{3}}$

Step 3.4.1.1.2.2

Rewrite the expression.

$y^{\frac{1}{2} \cdot 2} = {(\frac{9 x^{2}}{8} + \frac{3 K}{2})}^{\frac{2}{3}}$

Step 3.4.1.1.3

Cancel the common factor of $2$ .

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Step 3.4.1.1.3.1

Cancel the common factor.

$y^{\frac{1}{2} \cdot 2} = {(\frac{9 x^{2}}{8} + \frac{3 K}{2})}^{\frac{2}{3}}$

Step 3.4.1.1.3.2

Rewrite the expression.

$y^{1} = {(\frac{9 x^{2}}{8} + \frac{3 K}{2})}^{\frac{2}{3}}$

Step 3.4.1.2

Simplify.

$y = {(\frac{9 x^{2}}{8} + \frac{3 K}{2})}^{\frac{2}{3}}$

Step 4

Simplify the constant of integration.

$y = {(\frac{9 x^{2}}{8} + K)}^{\frac{2}{3}}$