Algebra Examples

$5^{- \frac{2}{3}} = \frac{125^{\frac{x}{3}}}{25^{\frac{4}{3}}}$

Step 1

Rewrite the equation as $\frac{125^{\frac{x}{3}}}{25^{\frac{4}{3}}} = 5^{- \frac{2}{3}}$ .

$\frac{125^{\frac{x}{3}}}{25^{\frac{4}{3}}} = 5^{- \frac{2}{3}}$

Step 2

Multiply both sides of the equation by $25^{\frac{4}{3}}$ .

$25^{\frac{4}{3}} \frac{125^{\frac{x}{3}}}{25^{\frac{4}{3}}} = 25^{\frac{4}{3}} \cdot 5^{- \frac{2}{3}}$

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

Cancel the common factor of $25^{\frac{4}{3}}$ .

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Step 3.1.1.1

Cancel the common factor.

$25^{\frac{4}{3}} \frac{125^{\frac{x}{3}}}{25^{\frac{4}{3}}} = 25^{\frac{4}{3}} \cdot 5^{- \frac{2}{3}}$

Step 3.1.1.2

Rewrite the expression.

$125^{\frac{x}{3}} = 25^{\frac{4}{3}} \cdot 5^{- \frac{2}{3}}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $25^{\frac{4}{3}} \cdot 5^{- \frac{2}{3}}$ .

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Step 3.2.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$125^{\frac{x}{3}} = 25^{\frac{4}{3}} \cdot \frac{1}{5^{\frac{2}{3}}}$

Step 3.2.1.2

Combine $25^{\frac{4}{3}}$ and $\frac{1}{5^{\frac{2}{3}}}$ .

$125^{\frac{x}{3}} = \frac{25^{\frac{4}{3}}}{5^{\frac{2}{3}}}$

Step 4

Move $5^{\frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$125^{\frac{x}{3}} = 25^{\frac{4}{3}} \cdot 5^{- \frac{2}{3}}$

Step 5

Rewrite $25$ as $5^{2}$ .

$125^{\frac{x}{3}} = {(5^{2})}^{\frac{4}{3}} \cdot 5^{- \frac{2}{3}}$

Step 6

Multiply the exponents in ${(5^{2})}^{\frac{4}{3}}$ .

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Step 6.1

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

$125^{\frac{x}{3}} = 5^{2 (\frac{4}{3})} \cdot 5^{- \frac{2}{3}}$

Step 6.2

Multiply $2 (\frac{4}{3})$ .

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Step 6.2.1

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

$125^{\frac{x}{3}} = 5^{\frac{2 \cdot 4}{3}} \cdot 5^{- \frac{2}{3}}$

Step 6.2.2

Multiply $2$ by $4$ .

$125^{\frac{x}{3}} = 5^{\frac{8}{3}} \cdot 5^{- \frac{2}{3}}$

Step 7

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

$125^{\frac{x}{3}} = 5^{\frac{8}{3} - \frac{2}{3}}$

Step 8

Combine the numerators over the common denominator.

$125^{\frac{x}{3}} = 5^{\frac{8 - 2}{3}}$

Step 9

Subtract $2$ from $8$ .

$125^{\frac{x}{3}} = 5^{\frac{6}{3}}$

Step 10

Cancel the common factor of $6$ and $3$ .

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Step 10.1

Factor $3$ out of $6$ .

$125^{\frac{x}{3}} = 5^{\frac{3 \cdot 2}{3}}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $3$ out of $3$ .

$125^{\frac{x}{3}} = 5^{\frac{3 \cdot 2}{3 (1)}}$

Step 10.2.2

Cancel the common factor.

$125^{\frac{x}{3}} = 5^{\frac{3 \cdot 2}{3 \cdot 1}}$

Step 10.2.3

Rewrite the expression.

$125^{\frac{x}{3}} = 5^{\frac{2}{1}}$

Step 10.2.4

Divide $2$ by $1$ .

$125^{\frac{x}{3}} = 5^{2}$

Step 11

Create equivalent expressions in the equation that all have equal bases.

$5^{3 \frac{x}{3}} = 5^{2}$

Step 12

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

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

Step 13

Cancel the common factor of $3$ .

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Step 13.1

Cancel the common factor.

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

Step 13.2

Rewrite the expression.

$x = 2$