Precalculus Examples

$2^{x} \cdot \frac{1}{32} = 32$

Step 1

Raise $32$ to the power of $1$ .

$2^{x} \cdot \frac{1}{32^{1}} = 32$

Step 2

Move $32^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$2^{x} \cdot 32^{- 1} = 32$

Step 3

Rewrite $32$ as $2^{5}$ .

$2^{x} \cdot {(2^{5})}^{- 1} = 32$

Step 4

Multiply the exponents in ${(2^{5})}^{- 1}$ .

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Step 4.1

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

$2^{x} \cdot 2^{5 \cdot - 1} = 32$

Step 4.2

Multiply $5$ by $- 1$ .

$2^{x} \cdot 2^{- 5} = 32$

Step 5

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

$2^{x - 5} = 32$

Step 6

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

$2^{x - 5} = 2^{5}$

Step 7

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

$x - 5 = 5$

Step 8

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

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Step 8.1

Add $5$ to both sides of the equation.

$x = 5 + 5$

Step 8.2

Add $5$ and $5$ .

$x = 10$