Precalculus Examples

\sqrt{\frac{4 x}{5}}

$\sqrt{\frac{4 x}{5}}$

Step 1

Rewrite

$\sqrt{\frac{4 x}{5}}$ as

$\frac{\sqrt{4 x}}{\sqrt{5}}$ .

$\frac{\sqrt{4 x}}{\sqrt{5}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite

$4$ as

$2^{2}$ .

$\frac{\sqrt{2^{2} x}}{\sqrt{5}}$

Step 2.2

Pull terms out from under the radical.

$\frac{2 \sqrt{x}}{\sqrt{5}}$

Step 3

Multiply

$\frac{2 \sqrt{x}}{\sqrt{5}}$ by

$\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{x}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply

$\frac{2 \sqrt{x}}{\sqrt{5}}$ by

$\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{2 \sqrt{x} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 4.2

Raise

$\sqrt{5}$ to the power of

$1$ .

$\frac{2 \sqrt{x} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 4.3

Raise

$\sqrt{5}$ to the power of

$1$ .

$\frac{2 \sqrt{x} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 4.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{2 \sqrt{x} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 4.5

Add

$1$ and

$1$ .

$\frac{2 \sqrt{x} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 4.6

Rewrite

${\sqrt{5}}^{2}$ as

$5$ .

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Step 4.6.1

Use

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

$\sqrt{5}$ as

$5^{\frac{1}{2}}$ .

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

Step 4.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 \sqrt{x} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{2 \sqrt{x} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of

$2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{x} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{x} \sqrt{5}}{5^{1}}$

Step 4.6.5

Evaluate the exponent.

$\frac{2 \sqrt{x} \sqrt{5}}{5}$

Step 5

Combine using the product rule for radicals.

$\frac{2 \sqrt{5 x}}{5}$

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