Calculus Examples

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

Step 1

Rewrite the equation as $\frac{1}{3} \cdot {(y^{2} + 2)}^{\frac{3}{2}} = x$ .

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

Step 2

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

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

Step 3

Simplify the left side.

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Step 3.1

Simplify ${(\frac{1}{3} \cdot {(y^{2} + 2)}^{\frac{3}{2}})}^{\frac{2}{3}}$ .

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Step 3.1.1

Combine fractions.

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Step 3.1.1.1

Combine $\frac{1}{3}$ and ${(y^{2} + 2)}^{\frac{3}{2}}$ .

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

Step 3.1.1.2

Apply the product rule to $\frac{{(y^{2} + 2)}^{\frac{3}{2}}}{3}$ .

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

Step 3.1.2

Simplify the numerator.

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Step 3.1.2.1

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

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Step 3.1.2.1.1

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

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

Step 3.1.2.1.2

Cancel the common factor of $3$ .

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Step 3.1.2.1.2.1

Cancel the common factor.

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

Step 3.1.2.1.2.2

Rewrite the expression.

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

Step 3.1.2.1.3

Cancel the common factor of $2$ .

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Step 3.1.2.1.3.1

Cancel the common factor.

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

Step 3.1.2.1.3.2

Rewrite the expression.

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

Step 3.1.2.2

Simplify.

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

Step 3.1.3

Split the fraction $\frac{y^{2} + 2}{3^{\frac{2}{3}}}$ into two fractions.

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

Step 4

Solve for $y$ .

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Step 4.1

Subtract $\frac{2}{3^{\frac{2}{3}}}$ from both sides of the equation.

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3^{\frac{2}{3}}$ .

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Step 4.2.2.1.1

Cancel the common factor.

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

Step 4.2.2.1.2

Rewrite the expression.

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

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $3^{\frac{2}{3}}$ .

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Step 4.2.3.1.1

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

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

Step 4.2.3.1.2

Cancel the common factor.

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

Step 4.2.3.1.3

Rewrite the expression.

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

Step 5

Combine $\frac{1}{3}$ and ${(y^{2} + 2)}^{\frac{3}{2}}$ .

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