Precalculus Examples

$8^{x} = {(\frac{1}{32})}^{x - 2}$

Step 1

Apply the product rule to $\frac{1}{32}$ .

$8^{x} = \frac{1^{x - 2}}{32^{x - 2}}$

Step 2

One to any power is one.

$8^{x} = \frac{1}{32^{x - 2}}$

Step 3

Move $32^{x - 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$8^{x} = 32^{- (x - 2)}$

Step 4

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

$2^{3 x} = 2^{5 (- (x - 2))}$

Step 5

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

$3 x = 5 (- (x - 2))$

Step 6

Solve for $x$ .

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Step 6.1

Simplify $5 (- (x - 2))$ .

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Step 6.1.1

Apply the distributive property.

$3 x = 5 (- x - - 2)$

Step 6.1.2

Multiply $- 1$ by $- 2$ .

$3 x = 5 (- x + 2)$

Step 6.1.3

Apply the distributive property.

$3 x = 5 (- x) + 5 \cdot 2$

Step 6.1.4

Multiply.

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Step 6.1.4.1

Multiply $- 1$ by $5$ .

$3 x = - 5 x + 5 \cdot 2$

Step 6.1.4.2

Multiply $5$ by $2$ .

$3 x = - 5 x + 10$

Step 6.2

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

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Step 6.2.1

Add $5 x$ to both sides of the equation.

$3 x + 5 x = 10$

Step 6.2.2

Add $3 x$ and $5 x$ .

$8 x = 10$

Step 6.3

Divide each term in $8 x = 10$ by $8$ and simplify.

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Step 6.3.1

Divide each term in $8 x = 10$ by $8$ .

$\frac{8 x}{8} = \frac{10}{8}$

Step 6.3.2

Simplify the left side.

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Step 6.3.2.1

Cancel the common factor of $8$ .

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Step 6.3.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{10}{8}$

Step 6.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{8}$

Step 6.3.3

Simplify the right side.

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Step 6.3.3.1

Cancel the common factor of $10$ and $8$ .

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Step 6.3.3.1.1

Factor $2$ out of $10$ .

$x = \frac{2 (5)}{8}$

Step 6.3.3.1.2

Cancel the common factors.

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Step 6.3.3.1.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 5}{2 \cdot 4}$

Step 6.3.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 5}{2 \cdot 4}$

Step 6.3.3.1.2.3

Rewrite the expression.

$x = \frac{5}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{4}$

Decimal Form:

$x = 1.25$

Mixed Number Form:

$x = 1 \frac{1}{4}$