Calculus Examples

\sqrt{\frac{8 x}{5}}

$\sqrt{\frac{8 x}{5}}$

Step 1

Rewrite

\sqrt{\frac{8 x}{5}}

$\sqrt{\frac{8 x}{5}}$ as

\frac{\sqrt{8 x}}{\sqrt{5}}

$\frac{\sqrt{8 x}}{\sqrt{5}}$ .

\frac{\sqrt{8 x}}{\sqrt{5}}

$\frac{\sqrt{8 x}}{\sqrt{5}}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite

8 x

$8 x$ as

2^{2} \cdot (2 x)

$2^{2} \cdot (2 x)$ .

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Step 2.1.1

Factor

4

$4$ out of

8

$8$ .

\frac{\sqrt{4 (2) x}}{\sqrt{5}}

$\frac{\sqrt{4 (2) x}}{\sqrt{5}}$

Step 2.1.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 2.1.3

Add parentheses.

\frac{\sqrt{2^{2} \cdot (2 x)}}{\sqrt{5}}

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

\frac{\sqrt{2^{2} \cdot (2 x)}}{\sqrt{5}}

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

Step 2.2

Pull terms out from under the radical.

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

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

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

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

Step 3

Multiply

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

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

\frac{\sqrt{5}}{\sqrt{5}}

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

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply

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

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

\frac{\sqrt{5}}{\sqrt{5}}

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

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

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

Step 4.2

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

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

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

Step 4.3

Raise

\sqrt{5}

$\sqrt{5}$ to the power of

1

$1$ .

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

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

Step 4.4

Use the power rule

a^{m} a^{n} = a^{m + n}

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

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

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

Step 4.5

Add

1

$1$ and

1

$1$ .

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

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

Step 4.6

Rewrite

{\sqrt{5}}^{2}

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

5

$5$ .

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Step 4.6.1

Use

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

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

\sqrt{5}

$\sqrt{5}$ as

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

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

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

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

Step 4.6.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 4.6.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 4.6.4

Cancel the common factor of

2

$2$ .

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Step 4.6.4.1

Cancel the common factor.

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

Step 4.6.4.2

Rewrite the expression.

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

Step 4.6.5

Evaluate the exponent.

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

Step 5

Simplify the numerator.

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Step 5.1

Combine using the product rule for radicals.

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

Step 5.2

Multiply

$5$ by

$2$ .

$\frac{2 \sqrt{10 x}}{5}$

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