Precalculus Examples

${(\frac{7}{8})}^{x} = 4^{1 - x}$

Step 1

Subtract $4^{1 - x}$ from both sides of the equation.

${(\frac{7}{8})}^{x} - 4^{1 - x} = 0$

Step 2

Simplify ${(\frac{7}{8})}^{x} - 4^{1 - x}$ .

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Step 2.1

Apply the product rule to $\frac{7}{8}$ .

$\frac{7^{x}}{8^{x}} - 4^{1 - x} = 0$

Step 2.2

To write $- 4^{1 - x}$ as a fraction with a common denominator, multiply by $\frac{8^{x}}{8^{x}}$ .

$\frac{7^{x}}{8^{x}} - 4^{1 - x} \cdot \frac{8^{x}}{8^{x}} = 0$

Step 2.3

Combine $- 4^{1 - x}$ and $\frac{8^{x}}{8^{x}}$ .

$\frac{7^{x}}{8^{x}} + \frac{- 4^{1 - x} \cdot 8^{x}}{8^{x}} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{7^{x} - 4^{1 - x} \cdot 8^{x}}{8^{x}} = 0$

Step 2.5

Multiply $- 4^{1 - x} \cdot 8^{x}$ .

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Step 2.5.1

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

$\frac{7^{x} - ({(2^{3})}^{x} \cdot 4^{1 - x})}{8^{x}} = 0$

Step 2.5.2

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

$\frac{7^{x} - (2^{3 x} \cdot 4^{1 - x})}{8^{x}} = 0$

Step 2.5.3

Rewrite $4$ as $2^{2}$ .

$\frac{7^{x} - (2^{3 x} \cdot {(2^{2})}^{1 - x})}{8^{x}} = 0$

Step 2.5.4

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

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Step 2.5.4.1

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

$\frac{7^{x} - (2^{3 x} \cdot 2^{2 (1 - x)})}{8^{x}} = 0$

Step 2.5.4.2

Apply the distributive property.

$\frac{7^{x} - (2^{3 x} \cdot 2^{2 \cdot 1 + 2 (- x)})}{8^{x}} = 0$

Step 2.5.4.3

Multiply $2$ by $1$ .

$\frac{7^{x} - (2^{3 x} \cdot 2^{2 + 2 (- x)})}{8^{x}} = 0$

Step 2.5.4.4

Multiply $- 1$ by $2$ .

$\frac{7^{x} - (2^{3 x} \cdot 2^{2 - 2 x})}{8^{x}} = 0$

Step 2.5.5

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

$\frac{7^{x} - 2^{3 x + 2 - 2 x}}{8^{x}} = 0$

Step 2.5.6

Subtract $2 x$ from $3 x$ .

$\frac{7^{x} - 2^{x + 2}}{8^{x}} = 0$