Precalculus Examples

$2^{x} \cdot 8^{- x} = 4^{x}$

Step 1

Rewrite $8$ as $2^{3}$ .

$2^{x} \cdot {(2^{3})}^{- x} = 4^{x}$

Step 2

Multiply the exponents in ${(2^{3})}^{- x}$ .

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Step 2.1

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

$2^{x} \cdot 2^{3 (- x)} = 4^{x}$

Step 2.2

Multiply $- 1$ by $3$ .

$2^{x} \cdot 2^{- 3 x} = 4^{x}$

Step 3

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

$2^{x - 3 x} = 4^{x}$

Step 4

Subtract $3 x$ from $x$ .

$2^{- 2 x} = 4^{x}$

Step 5

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

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

Step 6

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

$- 2 x = 2 x$

Step 7

Solve for $x$ .

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Step 7.1

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

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Step 7.1.1

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

$- 2 x - 2 x = 0$

Step 7.1.2

Subtract $2 x$ from $- 2 x$ .

$- 4 x = 0$

Step 7.2

Divide each term in $- 4 x = 0$ by $- 4$ and simplify.

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Step 7.2.1

Divide each term in $- 4 x = 0$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{0}{- 4}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $- 4$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{0}{- 4}$

Step 7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{- 4}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Divide $0$ by $- 4$ .

$x = 0$