Basic Math Examples

\sqrt{\frac{3}{\sqrt{60}}}

$\sqrt{\frac{3}{\sqrt{60}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite

$60$ as

$2^{2} \cdot 15$ .

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Step 1.1.1

Factor

$4$ out of

$60$ .

$\sqrt{\frac{3}{\sqrt{4 (15)}}}$

Step 1.1.2

Rewrite

$4$ as

$2^{2}$ .

$\sqrt{\frac{3}{\sqrt{2^{2} \cdot 15}}}$

Step 1.2

Pull terms out from under the radical.

$\sqrt{\frac{3}{2 \sqrt{15}}}$

Step 2

Multiply

$\frac{3}{2 \sqrt{15}}$ by

$\frac{\sqrt{15}}{\sqrt{15}}$ .

$\sqrt{\frac{3}{2 \sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply

$\frac{3}{2 \sqrt{15}}$ by

$\frac{\sqrt{15}}{\sqrt{15}}$ .

$\sqrt{\frac{3 \sqrt{15}}{2 \sqrt{15} \sqrt{15}}}$

Step 3.2

Move

$\sqrt{15}$ .

$\sqrt{\frac{3 \sqrt{15}}{2 (\sqrt{15} \sqrt{15})}}$

Step 3.3

Raise

$\sqrt{15}$ to the power of

$1$ .

$\sqrt{\frac{3 \sqrt{15}}{2 ({\sqrt{15}}^{1} \sqrt{15})}}$

Step 3.4

Raise

$\sqrt{15}$ to the power of

$1$ .

$\sqrt{\frac{3 \sqrt{15}}{2 ({\sqrt{15}}^{1} {\sqrt{15}}^{1})}}$

Step 3.5

Use the power rule

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

$\sqrt{\frac{3 \sqrt{15}}{2 {\sqrt{15}}^{1 + 1}}}$

Step 3.6

Add

$1$ and

$1$ .

$\sqrt{\frac{3 \sqrt{15}}{2 {\sqrt{15}}^{2}}}$

Step 3.7

Rewrite

${\sqrt{15}}^{2}$ as

$15$ .

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Step 3.7.1

Use

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

$\sqrt{15}$ as

$15^{\frac{1}{2}}$ .

$\sqrt{\frac{3 \sqrt{15}}{2 {(15^{\frac{1}{2}})}^{2}}}$

Step 3.7.2

Apply the power rule and multiply exponents,

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

$\sqrt{\frac{3 \sqrt{15}}{2 \cdot 15^{\frac{1}{2} \cdot 2}}}$

Step 3.7.3

Combine

$\frac{1}{2}$ and

$2$ .

$\sqrt{\frac{3 \sqrt{15}}{2 \cdot 15^{\frac{2}{2}}}}$

Step 3.7.4

Cancel the common factor of

$2$ .

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Step 3.7.4.1

Cancel the common factor.

$\sqrt{\frac{3 \sqrt{15}}{2 \cdot 15^{\frac{2}{2}}}}$

Step 3.7.4.2

Rewrite the expression.

$\sqrt{\frac{3 \sqrt{15}}{2 \cdot 15^{1}}}$

Step 3.7.5

Evaluate the exponent.

$\sqrt{\frac{3 \sqrt{15}}{2 \cdot 15}}$

Step 4

Cancel the common factor of

$3$ and

$15$ .

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Step 4.1

Factor

$3$ out of

$3 \sqrt{15}$ .

$\sqrt{\frac{3 (\sqrt{15})}{2 \cdot 15}}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor

$3$ out of

$2 \cdot 15$ .

$\sqrt{\frac{3 (\sqrt{15})}{3 (2 \cdot 5)}}$

Step 4.2.2

Cancel the common factor.

$\sqrt{\frac{3 \sqrt{15}}{3 (2 \cdot 5)}}$

Step 4.2.3

Rewrite the expression.

$\sqrt{\frac{\sqrt{15}}{2 \cdot 5}}$

Step 5

Multiply

$2$ by

$5$ .

$\sqrt{\frac{\sqrt{15}}{10}}$

Step 6

Rewrite

$\sqrt{\frac{\sqrt{15}}{10}}$ as

$\frac{\sqrt{\sqrt{15}}}{\sqrt{10}}$ .

$\frac{\sqrt{\sqrt{15}}}{\sqrt{10}}$

Step 7

Rewrite

$\sqrt{\sqrt{15}}$ as

$\sqrt[4]{15}$ .

$\frac{\sqrt[4]{15}}{\sqrt{10}}$

Step 8

Multiply

$\frac{\sqrt[4]{15}}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\sqrt[4]{15}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 9

Combine and simplify the denominator.

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Step 9.1

Multiply

$\frac{\sqrt[4]{15}}{\sqrt{10}}$ by

$\frac{\sqrt{10}}{\sqrt{10}}$ .

$\frac{\sqrt[4]{15} \sqrt{10}}{\sqrt{10} \sqrt{10}}$

Step 9.2

Raise

$\sqrt{10}$ to the power of

$1$ .

$\frac{\sqrt[4]{15} \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}}$

Step 9.3

Raise

$\sqrt{10}$ to the power of

$1$ .

$\frac{\sqrt[4]{15} \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}}$

Step 9.4

Use the power rule

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

$\frac{\sqrt[4]{15} \sqrt{10}}{{\sqrt{10}}^{1 + 1}}$

Step 9.5

Add

$1$ and

$1$ .

$\frac{\sqrt[4]{15} \sqrt{10}}{{\sqrt{10}}^{2}}$

Step 9.6

Rewrite

${\sqrt{10}}^{2}$ as

$10$ .

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Step 9.6.1

Use

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

$\sqrt{10}$ as

$10^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{15} \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}}$

Step 9.6.2

Apply the power rule and multiply exponents,

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

$\frac{\sqrt[4]{15} \sqrt{10}}{10^{\frac{1}{2} \cdot 2}}$

Step 9.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$\frac{\sqrt[4]{15} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 9.6.4

Cancel the common factor of

$2$ .

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Step 9.6.4.1

Cancel the common factor.

$\frac{\sqrt[4]{15} \sqrt{10}}{10^{\frac{2}{2}}}$

Step 9.6.4.2

Rewrite the expression.

$\frac{\sqrt[4]{15} \sqrt{10}}{10^{1}}$

Step 9.6.5

Evaluate the exponent.

$\frac{\sqrt[4]{15} \sqrt{10}}{10}$

Step 10

Simplify the numerator.

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Step 10.1

Rewrite the expression using the least common index of

$4$ .

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Step 10.1.1

Use

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

$\sqrt{10}$ as

$10^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{15} \cdot 10^{\frac{1}{2}}}{10}$

Step 10.1.2

Rewrite

$10^{\frac{1}{2}}$ as

$10^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{15} \cdot 10^{\frac{2}{4}}}{10}$

Step 10.1.3

Rewrite

$10^{\frac{2}{4}}$ as

$\sqrt[4]{10^{2}}$ .

$\frac{\sqrt[4]{15} \sqrt[4]{10^{2}}}{10}$

Step 10.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{15 \cdot 10^{2}}}{10}$

Step 11

Simplify the numerator.

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Step 11.1

Rewrite

$15 \cdot 10^{2}$ as

$0.15 \cdot 10^{4}$ .

$\frac{\sqrt[4]{0.15 \cdot 10^{4}}}{10}$

Step 11.2

Rewrite

$\sqrt[4]{0.15 \cdot 10^{4}}$ as

$\sqrt[4]{0.15} \cdot \sqrt[4]{10^{4}}$ .

$\frac{\sqrt[4]{0.15} \cdot \sqrt[4]{10^{4}}}{10}$

Step 11.3

Evaluate the root.

$\frac{0.62233297 \cdot \sqrt[4]{10^{4}}}{10}$

Step 11.4

Pull terms out from under the radical, assuming positive real numbers.

$\frac{0.62233297 \cdot 10}{10}$

Step 11.5

Raise

$10$ to the power of

$1$ .

$\frac{0.62233297 \cdot 10^{1}}{10}$

Step 11.6

Move the decimal point in

$0.62233297$ to the right by

$1$ place and decrease the power of

$10^{1}$ by

$1$ .

$\frac{6.22332977 \cdot 10^{0}}{10}$

Step 11.7

Convert

$6.22332977 \cdot 10^{0}$ from scientific notation.

$\frac{6.22332977}{10}$

Step 12

Divide

$6.22332977$ by

$10$ .

$0.62233297$