Algebra Examples

$125 = {(\frac{1}{25})}^{y - 1}$

Step 1

Rewrite the equation as ${(\frac{1}{25})}^{y - 1} = 125$ .

${(\frac{1}{25})}^{y - 1} = 125$

Step 2

Apply the product rule to $\frac{1}{25}$ .

$\frac{1^{y - 1}}{25^{y - 1}} = 125$

Step 3

One to any power is one.

$\frac{1}{25^{y - 1}} = 125$

Step 4

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

$25^{- (y - 1)} = 125$

Step 5

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

$5^{2 (- (y - 1))} = 5^{3}$

Step 6

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

$2 (- (y - 1)) = 3$

Step 7

Solve for $y$ .

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Step 7.1

Simplify.

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Step 7.1.1

Apply the distributive property.

$2 (- y - - 1) = 3$

Step 7.1.2

Multiply $- 1$ by $- 1$ .

$2 (- y + 1) = 3$

Step 7.2

Divide each term in $2 (- y + 1) = 3$ by $2$ and simplify.

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Step 7.2.1

Divide each term in $2 (- y + 1) = 3$ by $2$ .

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

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $2$ .

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Step 7.2.2.1.1

Cancel the common factor.

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

Step 7.2.2.1.2

Divide $- y + 1$ by $1$ .

$- y + 1 = \frac{3}{2}$

Step 7.3

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

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Step 7.3.1

Subtract $1$ from both sides of the equation.

$- y = \frac{3}{2} - 1$

Step 7.3.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.3.3

Combine $- 1$ and $\frac{2}{2}$ .

$- y = \frac{3}{2} + \frac{- 1 \cdot 2}{2}$

Step 7.3.4

Combine the numerators over the common denominator.

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

Step 7.3.5

Simplify the numerator.

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Step 7.3.5.1

Multiply $- 1$ by $2$ .

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

Step 7.3.5.2

Subtract $2$ from $3$ .

$- y = \frac{1}{2}$

Step 7.4

Divide each term in $- y = \frac{1}{2}$ by $- 1$ and simplify.

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Step 7.4.1

Divide each term in $- y = \frac{1}{2}$ by $- 1$ .

$\frac{- y}{- 1} = \frac{\frac{1}{2}}{- 1}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{\frac{1}{2}}{- 1}$

Step 7.4.2.2

Divide $y$ by $1$ .

$y = \frac{\frac{1}{2}}{- 1}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Move the negative one from the denominator of $\frac{\frac{1}{2}}{- 1}$ .

$y = - 1 \cdot \frac{1}{2}$

Step 7.4.3.2

Rewrite $- 1 \cdot \frac{1}{2}$ as $- \frac{1}{2}$ .

$y = - \frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{1}{2}$

Decimal Form:

$y = - 0.5$