Precalculus Examples

$\frac{\sqrt[9]{125}}{25^{- 1}} = 25^{2 - w}$

Step 1

Rewrite the equation as $25^{2 - w} = \frac{\sqrt[9]{125}}{25^{- 1}}$ .

$25^{2 - w} = \frac{\sqrt[9]{125}}{25^{- 1}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[9]{125}$ as $125^{\frac{1}{9}}$ .

$25^{2 - w} = \frac{125^{\frac{1}{9}}}{25^{- 1}}$

Step 3

Move $25^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$25^{2 - w} = 125^{\frac{1}{9}} \cdot 25$

Step 4

Rewrite $125$ as $5^{3}$ .

$25^{2 - w} = {(5^{3})}^{\frac{1}{9}} \cdot 25$

Step 5

Multiply the exponents in ${(5^{3})}^{\frac{1}{9}}$ .

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Step 5.1

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

$25^{2 - w} = 5^{3 (\frac{1}{9})} \cdot 25$

Step 5.2

Cancel the common factor of $3$ .

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Step 5.2.1

Factor $3$ out of $9$ .

$25^{2 - w} = 5^{3 \frac{1}{3 (3)}} \cdot 25$

Step 5.2.2

Cancel the common factor.

$25^{2 - w} = 5^{3 \frac{1}{3 \cdot 3}} \cdot 25$

Step 5.2.3

Rewrite the expression.

$25^{2 - w} = 5^{\frac{1}{3}} \cdot 25$

Step 6

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

$25^{2 - w} = 5^{\frac{1}{3}} \cdot 5^{2}$

Step 7

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

$25^{2 - w} = 5^{\frac{1}{3} + 2}$

Step 8

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$25^{2 - w} = 5^{\frac{1}{3} + 2 \cdot \frac{3}{3}}$

Step 9

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

$25^{2 - w} = 5^{\frac{1}{3} + \frac{2 \cdot 3}{3}}$

Step 10

Combine the numerators over the common denominator.

$25^{2 - w} = 5^{\frac{1 + 2 \cdot 3}{3}}$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $2$ by $3$ .

$25^{2 - w} = 5^{\frac{1 + 6}{3}}$

Step 11.2

Add $1$ and $6$ .

$25^{2 - w} = 5^{\frac{7}{3}}$

Step 12

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

$5^{2 (2 - w)} = 5^{\frac{7}{3}}$

Step 13

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

$2 (2 - w) = \frac{7}{3}$

Step 14

Solve for $w$ .

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Step 14.1

Divide each term in $2 (2 - w) = \frac{7}{3}$ by $2$ and simplify.

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Step 14.1.1

Divide each term in $2 (2 - w) = \frac{7}{3}$ by $2$ .

$\frac{2 (2 - w)}{2} = \frac{\frac{7}{3}}{2}$

Step 14.1.2

Simplify the left side.

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Step 14.1.2.1

Cancel the common factor of $2$ .

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Step 14.1.2.1.1

Cancel the common factor.

$\frac{2 (2 - w)}{2} = \frac{\frac{7}{3}}{2}$

Step 14.1.2.1.2

Divide $2 - w$ by $1$ .

$2 - w = \frac{\frac{7}{3}}{2}$

Step 14.1.3

Simplify the right side.

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Step 14.1.3.1

Multiply the numerator by the reciprocal of the denominator.

$2 - w = \frac{7}{3} \cdot \frac{1}{2}$

Step 14.1.3.2

Multiply $\frac{7}{3} \cdot \frac{1}{2}$ .

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Step 14.1.3.2.1

Multiply $\frac{7}{3}$ by $\frac{1}{2}$ .

$2 - w = \frac{7}{3 \cdot 2}$

Step 14.1.3.2.2

Multiply $3$ by $2$ .

$2 - w = \frac{7}{6}$

Step 14.2

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

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Step 14.2.1

Subtract $2$ from both sides of the equation.

$- w = \frac{7}{6} - 2$

Step 14.2.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$- w = \frac{7}{6} - 2 \cdot \frac{6}{6}$

Step 14.2.3

Combine $- 2$ and $\frac{6}{6}$ .

$- w = \frac{7}{6} + \frac{- 2 \cdot 6}{6}$

Step 14.2.4

Combine the numerators over the common denominator.

$- w = \frac{7 - 2 \cdot 6}{6}$

Step 14.2.5

Simplify the numerator.

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Step 14.2.5.1

Multiply $- 2$ by $6$ .

$- w = \frac{7 - 12}{6}$

Step 14.2.5.2

Subtract $12$ from $7$ .

$- w = \frac{- 5}{6}$

Step 14.2.6

Move the negative in front of the fraction.

$- w = - \frac{5}{6}$

Step 14.3

Divide each term in $- w = - \frac{5}{6}$ by $- 1$ and simplify.

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Step 14.3.1

Divide each term in $- w = - \frac{5}{6}$ by $- 1$ .

$\frac{- w}{- 1} = \frac{- \frac{5}{6}}{- 1}$

Step 14.3.2

Simplify the left side.

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Step 14.3.2.1

Dividing two negative values results in a positive value.

$\frac{w}{1} = \frac{- \frac{5}{6}}{- 1}$

Step 14.3.2.2

Divide $w$ by $1$ .

$w = \frac{- \frac{5}{6}}{- 1}$

Step 14.3.3

Simplify the right side.

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Step 14.3.3.1

Dividing two negative values results in a positive value.

$w = \frac{\frac{5}{6}}{1}$

Step 14.3.3.2

Divide $\frac{5}{6}$ by $1$ .

$w = \frac{5}{6}$

Step 15

The result can be shown in multiple forms.

Exact Form:

$w = \frac{5}{6}$

Decimal Form:

$w = 0.8\overline{3}$