Precalculus Examples

$\sqrt{2} = 4^{x}$

Step 1

Rewrite the equation as $4^{x} = \sqrt{2}$ .

$4^{x} = \sqrt{2}$

Step 2

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

$4^{x} = 2^{\frac{1}{2}}$

Step 3

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

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

Step 4

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

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

Step 5

Divide each term in $2 x = \frac{1}{2}$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 x = \frac{1}{2}$ by $2$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $x$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 5.3.2

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

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Step 5.3.2.1

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

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

Step 5.3.2.2

Multiply $2$ by $2$ .

$x = \frac{1}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{4}$

Decimal Form:

$x = 0.25$