Pre-Algebra Examples

$y = - \frac{7}{8} x^{3}$

Step 1

Rewrite the equation as $- \frac{7}{8} x^{3} = y$ .

$- \frac{7}{8} x^{3} = y$

Step 2

Multiply both sides of the equation by $- \frac{8}{7}$ .

$- \frac{8}{7} (- \frac{7}{8} x^{3}) = - \frac{8}{7} y$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $- \frac{8}{7} (- \frac{7}{8} x^{3})$ .

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Step 3.1.1.1

Combine $x^{3}$ and $\frac{7}{8}$ .

$- \frac{8}{7} (- \frac{x^{3} \cdot 7}{8}) = - \frac{8}{7} y$

Step 3.1.1.2

Cancel the common factor of $8$ .

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Step 3.1.1.2.1

Move the leading negative in $- \frac{8}{7}$ into the numerator.

$\frac{- 8}{7} (- \frac{x^{3} \cdot 7}{8}) = - \frac{8}{7} y$

Step 3.1.1.2.2

Move the leading negative in $- \frac{x^{3} \cdot 7}{8}$ into the numerator.

$\frac{- 8}{7} \cdot \frac{- x^{3} \cdot 7}{8} = - \frac{8}{7} y$

Step 3.1.1.2.3

Factor $8$ out of $- 8$ .

$\frac{8 (- 1)}{7} \cdot \frac{- x^{3} \cdot 7}{8} = - \frac{8}{7} y$

Step 3.1.1.2.4

Cancel the common factor.

$\frac{8 \cdot - 1}{7} \cdot \frac{- x^{3} \cdot 7}{8} = - \frac{8}{7} y$

Step 3.1.1.2.5

Rewrite the expression.

$\frac{- 1}{7} (- x^{3} \cdot 7) = - \frac{8}{7} y$

Step 3.1.1.3

Cancel the common factor of $7$ .

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Step 3.1.1.3.1

Factor $7$ out of $- x^{3} \cdot 7$ .

$\frac{- 1}{7} (7 \cdot (- x^{3})) = - \frac{8}{7} y$

Step 3.1.1.3.2

Cancel the common factor.

$\frac{- 1}{7} (7 \cdot (- x^{3})) = - \frac{8}{7} y$

Step 3.1.1.3.3

Rewrite the expression.

$- - x^{3} = - \frac{8}{7} y$

Step 3.1.1.4

Multiply.

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Step 3.1.1.4.1

Multiply $- 1$ by $- 1$ .

$1 x^{3} = - \frac{8}{7} y$

Step 3.1.1.4.2

Multiply $x^{3}$ by $1$ .

$x^{3} = - \frac{8}{7} y$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $- \frac{8}{7} y$ .

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Step 3.2.1.1

Combine $y$ and $\frac{8}{7}$ .

$x^{3} = - \frac{y \cdot 8}{7}$

Step 3.2.1.2

Move $8$ to the left of $y$ .

$x^{3} = - \frac{8 y}{7}$

Step 4

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

$x = \sqrt[3]{- \frac{8 y}{7}}$

Step 5

Simplify $\sqrt[3]{- \frac{8 y}{7}}$ .

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Step 5.1

Rewrite $- \frac{8 y}{7}$ as ${({(- 1)}^{3})}^{3} \frac{8 y}{7}$ .

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Step 5.1.1

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \frac{8 y}{7}}$

Step 5.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{({(- 1)}^{3})}^{3} \frac{8 y}{7}}$

Step 5.2

Pull terms out from under the radical.

$x = {(- 1)}^{3} \sqrt[3]{\frac{8 y}{7}}$

Step 5.3

Raise $- 1$ to the power of $3$ .

$x = - \sqrt[3]{\frac{8 y}{7}}$

Step 5.4

Rewrite $\sqrt[3]{\frac{8 y}{7}}$ as $\frac{\sqrt[3]{8 y}}{\sqrt[3]{7}}$ .

$x = - \frac{\sqrt[3]{8 y}}{\sqrt[3]{7}}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Rewrite $8$ as $2^{3}$ .

$x = - \frac{\sqrt[3]{2^{3} y}}{\sqrt[3]{7}}$

Step 5.5.2

Pull terms out from under the radical.

$x = - \frac{2 \sqrt[3]{y}}{\sqrt[3]{7}}$

Step 5.6

Multiply $\frac{2 \sqrt[3]{y}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$x = - (\frac{2 \sqrt[3]{y}}{\sqrt[3]{7}} \cdot \frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}})$

Step 5.7

Combine and simplify the denominator.

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Step 5.7.1

Multiply $\frac{2 \sqrt[3]{y}}{\sqrt[3]{7}}$ by $\frac{{\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{2}}$ .

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{\sqrt[3]{7} {\sqrt[3]{7}}^{2}}$

Step 5.7.2

Raise $\sqrt[3]{7}$ to the power of $1$ .

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1} {\sqrt[3]{7}}^{2}}$

Step 5.7.3

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

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{1 + 2}}$

Step 5.7.4

Add $1$ and $2$ .

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{{\sqrt[3]{7}}^{3}}$

Step 5.7.5

Rewrite ${\sqrt[3]{7}}^{3}$ as $7$ .

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Step 5.7.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{7}$ as $7^{\frac{1}{3}}$ .

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{{(7^{\frac{1}{3}})}^{3}}$

Step 5.7.5.2

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

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{7^{\frac{1}{3} \cdot 3}}$

Step 5.7.5.3

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

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 5.7.5.4

Cancel the common factor of $3$ .

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Step 5.7.5.4.1

Cancel the common factor.

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{7^{\frac{3}{3}}}$

Step 5.7.5.4.2

Rewrite the expression.

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{7^{1}}$

Step 5.7.5.5

Evaluate the exponent.

$x = - \frac{2 \sqrt[3]{y} {\sqrt[3]{7}}^{2}}{7}$

Step 5.8

Simplify the numerator.

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Step 5.8.1

Rewrite ${\sqrt[3]{7}}^{2}$ as $\sqrt[3]{7^{2}}$ .

$x = - \frac{2 \sqrt[3]{y} \sqrt[3]{7^{2}}}{7}$

Step 5.8.2

Raise $7$ to the power of $2$ .

$x = - \frac{2 \sqrt[3]{y} \sqrt[3]{49}}{7}$

Step 5.8.3

Combine using the product rule for radicals.

$x = - \frac{2 \sqrt[3]{49 y}}{7}$