Precalculus Examples

$\frac{1}{\sqrt{8}} = 4^{(m + 2)}$

Step 1

Rewrite the equation as $4^{m + 2} = \frac{1}{\sqrt{8}}$ .

$4^{m + 2} = \frac{1}{\sqrt{8}}$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8}$ as $8^{\frac{1}{2}}$ .

$4^{m + 2} = \frac{1}{8^{\frac{1}{2}}}$

Step 3

Move $8^{\frac{1}{2}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$4^{m + 2} = 8^{- \frac{1}{2}}$

Step 4

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

$2^{2 (m + 2)} = 2^{3 (- \frac{1}{2})}$

Step 5

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

$2 (m + 2) = 3 (- \frac{1}{2})$

Step 6

Solve for $m$ .

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Step 6.1

Simplify.

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Step 6.1.1

Multiply $3 (- \frac{1}{2})$ .

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Step 6.1.1.1

Multiply $- 1$ by $3$ .

$2 (m + 2) = - 3 (\frac{1}{2})$

Step 6.1.1.2

Combine $- 3$ and $\frac{1}{2}$ .

$2 (m + 2) = \frac{- 3}{2}$

Step 6.1.2

Move the negative in front of the fraction.

$2 (m + 2) = - \frac{3}{2}$

Step 6.2

Divide each term in $2 (m + 2) = - \frac{3}{2}$ by $2$ and simplify.

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Step 6.2.1

Divide each term in $2 (m + 2) = - \frac{3}{2}$ by $2$ .

$\frac{2 (m + 2)}{2} = \frac{- \frac{3}{2}}{2}$

Step 6.2.2

Simplify the left side.

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Step 6.2.2.1

Cancel the common factor of $2$ .

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Step 6.2.2.1.1

Cancel the common factor.

$\frac{2 (m + 2)}{2} = \frac{- \frac{3}{2}}{2}$

Step 6.2.2.1.2

Divide $m + 2$ by $1$ .

$m + 2 = \frac{- \frac{3}{2}}{2}$

Step 6.2.3

Simplify the right side.

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Step 6.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$m + 2 = - \frac{3}{2} \cdot \frac{1}{2}$

Step 6.2.3.2

Multiply $- \frac{3}{2} \cdot \frac{1}{2}$ .

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Step 6.2.3.2.1

Multiply $\frac{1}{2}$ by $\frac{3}{2}$ .

$m + 2 = - \frac{3}{2 \cdot 2}$

Step 6.2.3.2.2

Multiply $2$ by $2$ .

$m + 2 = - \frac{3}{4}$

Step 6.3

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

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Step 6.3.1

Subtract $2$ from both sides of the equation.

$m = - \frac{3}{4} - 2$

Step 6.3.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$m = - \frac{3}{4} - 2 \cdot \frac{4}{4}$

Step 6.3.3

Combine $- 2$ and $\frac{4}{4}$ .

$m = - \frac{3}{4} + \frac{- 2 \cdot 4}{4}$

Step 6.3.4

Combine the numerators over the common denominator.

$m = \frac{- 3 - 2 \cdot 4}{4}$

Step 6.3.5

Simplify the numerator.

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Step 6.3.5.1

Multiply $- 2$ by $4$ .

$m = \frac{- 3 - 8}{4}$

Step 6.3.5.2

Subtract $8$ from $- 3$ .

$m = \frac{- 11}{4}$

Step 6.3.6

Move the negative in front of the fraction.

$m = - \frac{11}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$m = - \frac{11}{4}$

Decimal Form:

$m = - 2.75$

Mixed Number Form:

$m = - 2 \frac{3}{4}$