Precalculus Examples

$16^{\frac{1}{3}} = {(\frac{1}{64})}^{3 m}$

Step 1

Rewrite the equation as ${(\frac{1}{64})}^{3 m} = 16^{\frac{1}{3}}$ .

${(\frac{1}{64})}^{3 m} = 16^{\frac{1}{3}}$

Step 2

Apply the product rule to $\frac{1}{64}$ .

$\frac{1^{3 m}}{64^{3 m}} = 16^{\frac{1}{3}}$

Step 3

One to any power is one.

$\frac{1}{64^{3 m}} = 16^{\frac{1}{3}}$

Step 4

Move $64^{3 m}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

$64^{- 3 m} = 16^{\frac{1}{3}}$

Step 5

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

$4^{3 (- 3 m)} = 4^{2 (\frac{1}{3})}$

Step 6

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

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

Step 7

Solve for $m$ .

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Step 7.1

Simplify.

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Step 7.1.1

Multiply $- 3$ by $3$ .

$- 9 m = 2 (\frac{1}{3})$

Step 7.1.2

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

$- 9 m = \frac{2}{3}$

Step 7.2

Divide each term in $- 9 m = \frac{2}{3}$ by $- 9$ and simplify.

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Step 7.2.1

Divide each term in $- 9 m = \frac{2}{3}$ by $- 9$ .

$\frac{- 9 m}{- 9} = \frac{\frac{2}{3}}{- 9}$

Step 7.2.2

Simplify the left side.

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Step 7.2.2.1

Cancel the common factor of $- 9$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{- 9 m}{- 9} = \frac{\frac{2}{3}}{- 9}$

Step 7.2.2.1.2

Divide $m$ by $1$ .

$m = \frac{\frac{2}{3}}{- 9}$

Step 7.2.3

Simplify the right side.

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Step 7.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$m = \frac{2}{3} \cdot \frac{1}{- 9}$

Step 7.2.3.2

Move the negative in front of the fraction.

$m = \frac{2}{3} (- \frac{1}{9})$

Step 7.2.3.3

Multiply $\frac{2}{3} (- \frac{1}{9})$ .

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Step 7.2.3.3.1

Multiply $\frac{2}{3}$ by $\frac{1}{9}$ .

$m = - \frac{2}{3 \cdot 9}$

Step 7.2.3.3.2

Multiply $3$ by $9$ .

$m = - \frac{2}{27}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$m = - \frac{2}{27}$

Decimal Form:

$m = - 0.\overline{074}$