Precalculus Examples

$4^{x} = \sqrt[7]{\frac{1}{16}}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[7]{\frac{1}{16}}$ as ${(\frac{1}{16})}^{\frac{1}{7}}$ .

$4^{x} = {(\frac{1}{16})}^{\frac{1}{7}}$

Step 2

Apply the product rule to $\frac{1}{16}$ .

$4^{x} = \frac{1^{\frac{1}{7}}}{16^{\frac{1}{7}}}$

Step 3

One to any power is one.

$4^{x} = \frac{1}{16^{\frac{1}{7}}}$

Step 4

Move $16^{\frac{1}{7}}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$4^{x} = 16^{- \frac{1}{7}}$

Step 5

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

$4^{x} = 4^{2 (- \frac{1}{7})}$

Step 6

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

$x = 2 (- \frac{1}{7})$

Step 7

Simplify $2 (- \frac{1}{7})$ .

Tap for more steps...

Step 7.1

Multiply $2 (- \frac{1}{7})$ .

Tap for more steps...

Step 7.1.1

Multiply $- 1$ by $2$ .

$x = - 2 (\frac{1}{7})$

Step 7.1.2

Combine $- 2$ and $\frac{1}{7}$ .

$x = \frac{- 2}{7}$

Step 7.2

Move the negative in front of the fraction.

$x = - \frac{2}{7}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{2}{7}$

Decimal Form:

$x = - 0.\overline{285714}$