Precalculus Examples

$\frac{1}{10^{y}} = 100^{y + 3}$

Step 1

Move $10^{y}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$10^{- y} = 100^{y + 3}$

Step 2

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

$10^{- y} = 10^{2 (y + 3)}$

Step 3

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

$- y = 2 (y + 3)$

Step 4

Solve for $y$ .

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Step 4.1

Simplify $2 (y + 3)$ .

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Step 4.1.1

Apply the distributive property.

$- y = 2 y + 2 \cdot 3$

Step 4.1.2

Multiply $2$ by $3$ .

$- y = 2 y + 6$

Step 4.2

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

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Step 4.2.1

Subtract $2 y$ from both sides of the equation.

$- y - 2 y = 6$

Step 4.2.2

Subtract $2 y$ from $- y$ .

$- 3 y = 6$

Step 4.3

Divide each term in $- 3 y = 6$ by $- 3$ and simplify.

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Step 4.3.1

Divide each term in $- 3 y = 6$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{6}{- 3}$

Step 4.3.2

Simplify the left side.

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Step 4.3.2.1

Cancel the common factor of $- 3$ .

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Step 4.3.2.1.1

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{6}{- 3}$

Step 4.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{- 3}$

Step 4.3.3

Simplify the right side.

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Step 4.3.3.1

Divide $6$ by $- 3$ .

$y = - 2$