Algebra Examples

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

Step 1

Set $- \frac{1}{8} \cdot {(x - 7)}^{3} - 8$ equal to $0$ .

$- \frac{1}{8} \cdot {(x - 7)}^{3} - 8 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Add $8$ to both sides of the equation.

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

Step 2.2

Combine ${(x - 7)}^{3}$ and $\frac{1}{8}$ .

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

Step 2.3

Multiply both sides of the equation by $- 8$ .

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

Step 2.4

Simplify both sides of the equation.

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Step 2.4.1

Simplify the left side.

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Step 2.4.1.1

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

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Step 2.4.1.1.1

Cancel the common factor of $8$ .

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Step 2.4.1.1.1.1

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

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

Step 2.4.1.1.1.2

Factor $8$ out of $- 8$ .

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

Step 2.4.1.1.1.3

Cancel the common factor.

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

Step 2.4.1.1.1.4

Rewrite the expression.

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

Step 2.4.1.1.2

Multiply.

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Step 2.4.1.1.2.1

Multiply $- 1$ by $- 1$ .

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

Step 2.4.1.1.2.2

Multiply ${(x - 7)}^{3}$ by $1$ .

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

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $- 8$ by $8$ .

${(x - 7)}^{3} = - 64$

Step 2.5

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

$x - 7 = \sqrt[3]{- 64}$

Step 2.6

Simplify $\sqrt[3]{- 64}$ .

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Step 2.6.1

Rewrite $- 64$ as ${(- 4)}^{3}$ .

$x - 7 = \sqrt[3]{{(- 4)}^{3}}$

Step 2.6.2

Pull terms out from under the radical, assuming real numbers.

$x - 7 = - 4$

Step 2.7

Move all terms not containing $x$ to the right side of the equation.

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Step 2.7.1

Add $7$ to both sides of the equation.

$x = - 4 + 7$

Step 2.7.2

Add $- 4$ and $7$ .

$x = 3$

Step 2.8

The multiplicity of a root is the number of times the root appears. For example, a factor of ${(x + 5)}^{3}$ would have a root at $x = - 5$ with multiplicity of $3$ .

$x = 3$ (Multiplicity of $3$ )

Step 3