Precalculus Examples

\sqrt{125} = 5^{x}

$\sqrt{125} = 5^{x}$

Step 1

Rewrite the equation as

5^{x} = \sqrt{125}

$5^{x} = \sqrt{125}$ .

5^{x} = \sqrt{125}

$5^{x} = \sqrt{125}$

Step 2

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{125}

$\sqrt{125}$ as

125^{\frac{1}{2}}

$125^{\frac{1}{2}}$ .

5^{x} = 125^{\frac{1}{2}}

$5^{x} = 125^{\frac{1}{2}}$

Step 3

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

5^{x} = 5^{3 (\frac{1}{2})}

$5^{x} = 5^{3 (\frac{1}{2})}$

Step 4

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

x = 3 (\frac{1}{2})

$x = 3 (\frac{1}{2})$

Step 5

Combine

3

$3$ and

\frac{1}{2}

$\frac{1}{2}$ .

x = \frac{3}{2}

$x = \frac{3}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

x = \frac{3}{2}

$x = \frac{3}{2}$

Decimal Form:

x = 1.5

$x = 1.5$

Mixed Number Form:

x = 1 \frac{1}{2}

$x = 1 \frac{1}{2}$