Algebra Examples

${(\sqrt[3]{5})}^{- x} = {(\frac{1}{5})}^{x + 2}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{5}$ as $5^{\frac{1}{3}}$ .

${(5^{\frac{1}{3}})}^{- x} = {(\frac{1}{5})}^{x + 2}$

Step 2

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

$5^{\frac{1}{3} (- x)} = {(\frac{1}{5})}^{x + 2}$

Step 3

Apply the product rule to $\frac{1}{5}$ .

$5^{\frac{1}{3} (- x)} = \frac{1^{x + 2}}{5^{x + 2}}$

Step 4

One to any power is one.

$5^{\frac{1}{3} (- x)} = \frac{1}{5^{x + 2}}$

Step 5

Move $5^{x + 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$5^{\frac{1}{3} (- x)} = 5^{- (x + 2)}$

Step 6

Since the bases are the same, then two expressions are only equal if the exponents are also equal.

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

Step 7

Solve for $x$ .

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Step 7.1

Simplify $\frac{1}{3} \cdot (- 1 x)$ .

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Step 7.1.1

Rewrite.

$0 + 0 + \frac{1}{3} \cdot (- 1 x) = - (x + 2)$

Step 7.1.2

Rewrite $- 1 x$ as $- x$ .

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

Step 7.1.3

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

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

Step 7.2

Simplify $- (x + 2)$ .

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Step 7.2.1

Apply the distributive property.

$- \frac{x}{3} = - x - 1 \cdot 2$

Step 7.2.2

Multiply $- 1$ by $2$ .

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

Step 7.3

Move all terms containing $x$ to the left side of the equation.

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Step 7.3.1

Add $x$ to both sides of the equation.

$- \frac{x}{3} + x = - 2$

Step 7.3.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7.3.3

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

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

Step 7.3.4

Combine the numerators over the common denominator.

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

Step 7.3.5

Simplify the numerator.

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Step 7.3.5.1

Move $3$ to the left of $x$ .

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

Step 7.3.5.2

Add $- x$ and $3 x$ .

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

Step 7.4

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

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

Step 7.5

Simplify both sides of the equation.

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Step 7.5.1

Simplify the left side.

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Step 7.5.1.1

Simplify $\frac{3}{2} \cdot \frac{2 x}{3}$ .

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Step 7.5.1.1.1

Cancel the common factor of $3$ .

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Step 7.5.1.1.1.1

Cancel the common factor.

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

Step 7.5.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 x) = \frac{3}{2} \cdot - 2$

Step 7.5.1.1.2

Cancel the common factor of $2$ .

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Step 7.5.1.1.2.1

Factor $2$ out of $2 x$ .

$\frac{1}{2} (2 (x)) = \frac{3}{2} \cdot - 2$

Step 7.5.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 x) = \frac{3}{2} \cdot - 2$

Step 7.5.1.1.2.3

Rewrite the expression.

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

Step 7.5.2

Simplify the right side.

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Step 7.5.2.1

Simplify $\frac{3}{2} \cdot - 2$ .

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Step 7.5.2.1.1

Cancel the common factor of $2$ .

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Step 7.5.2.1.1.1

Factor $2$ out of $- 2$ .

$x = \frac{3}{2} \cdot (2 (- 1))$

Step 7.5.2.1.1.2

Cancel the common factor.

$x = \frac{3}{2} \cdot (2 \cdot - 1)$

Step 7.5.2.1.1.3

Rewrite the expression.

$x = 3 \cdot - 1$

Step 7.5.2.1.2

Multiply $3$ by $- 1$ .

$x = - 3$