Pre-Algebra Examples

$\sqrt{\frac{3}{\sqrt{75}}}$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 1.1.1

Factor $25$ out of $75$ .

$\sqrt{\frac{3}{\sqrt{25 (3)}}}$

Step 1.1.2

Rewrite $25$ as $5^{2}$ .

$\sqrt{\frac{3}{\sqrt{5^{2} \cdot 3}}}$

Step 1.2

Pull terms out from under the radical.

$\sqrt{\frac{3}{5 \sqrt{3}}}$

Step 2

Multiply $\frac{3}{5 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{\frac{3}{5 \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}}$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{3}{5 \sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\sqrt{\frac{3 \sqrt{3}}{5 \sqrt{3} \sqrt{3}}}$

Step 3.2

Move $\sqrt{3}$ .

$\sqrt{\frac{3 \sqrt{3}}{5 (\sqrt{3} \sqrt{3})}}$

Step 3.3

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{\frac{3 \sqrt{3}}{5 ({\sqrt{3}}^{1} \sqrt{3})}}$

Step 3.4

Raise $\sqrt{3}$ to the power of $1$ .

$\sqrt{\frac{3 \sqrt{3}}{5 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})}}$

Step 3.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sqrt{\frac{3 \sqrt{3}}{5 {\sqrt{3}}^{1 + 1}}}$

Step 3.6

Add $1$ and $1$ .

$\sqrt{\frac{3 \sqrt{3}}{5 {\sqrt{3}}^{2}}}$

Step 3.7

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

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

Step 3.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\sqrt{\frac{3 \sqrt{3}}{5 \cdot 3^{\frac{1}{2} \cdot 2}}}$

Step 3.7.3

Combine $\frac{1}{2}$ and $2$ .

$\sqrt{\frac{3 \sqrt{3}}{5 \cdot 3^{\frac{2}{2}}}}$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\sqrt{\frac{3 \sqrt{3}}{5 \cdot 3^{\frac{2}{2}}}}$

Step 3.7.4.2

Rewrite the expression.

$\sqrt{\frac{3 \sqrt{3}}{5 \cdot 3^{1}}}$

Step 3.7.5

Evaluate the exponent.

$\sqrt{\frac{3 \sqrt{3}}{5 \cdot 3}}$

Step 4

Cancel the common factor of $3$ .

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Step 4.1

Cancel the common factor.

$\sqrt{\frac{3 \sqrt{3}}{5 \cdot 3}}$

Step 4.2

Rewrite the expression.

$\sqrt{\frac{\sqrt{3}}{5}}$

Step 5

Rewrite $\sqrt{\frac{\sqrt{3}}{5}}$ as $\frac{\sqrt{\sqrt{3}}}{\sqrt{5}}$ .

$\frac{\sqrt{\sqrt{3}}}{\sqrt{5}}$

Step 6

Rewrite $\sqrt{\sqrt{3}}$ as $\sqrt[4]{3}$ .

$\frac{\sqrt[4]{3}}{\sqrt{5}}$

Step 7

Multiply $\frac{\sqrt[4]{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt[4]{3}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{\sqrt[4]{3}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{\sqrt[4]{3} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 8.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt[4]{3} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 8.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{\sqrt[4]{3} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 8.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sqrt[4]{3} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{\sqrt[4]{3} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 8.6

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

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Step 8.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{3} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 8.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{\sqrt[4]{3} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{\sqrt[4]{3} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{\sqrt[4]{3} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{\sqrt[4]{3} \sqrt{5}}{5^{1}}$

Step 8.6.5

Evaluate the exponent.

$\frac{\sqrt[4]{3} \sqrt{5}}{5}$

Step 9

Simplify the numerator.

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Step 9.1

Rewrite the expression using the least common index of $4$ .

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Step 9.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{\sqrt[4]{3} \cdot 5^{\frac{1}{2}}}{5}$

Step 9.1.2

Rewrite $5^{\frac{1}{2}}$ as $5^{\frac{2}{4}}$ .

$\frac{\sqrt[4]{3} \cdot 5^{\frac{2}{4}}}{5}$

Step 9.1.3

Rewrite $5^{\frac{2}{4}}$ as $\sqrt[4]{5^{2}}$ .

$\frac{\sqrt[4]{3} \sqrt[4]{5^{2}}}{5}$

Step 9.2

Combine using the product rule for radicals.

$\frac{\sqrt[4]{3 \cdot 5^{2}}}{5}$

Step 9.3

Raise $5$ to the power of $2$ .

$\frac{\sqrt[4]{3 \cdot 25}}{5}$

Step 10

Multiply $3$ by $25$ .

$\frac{\sqrt[4]{75}}{5}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{\sqrt[4]{75}}{5}$

Decimal Form:

$0.58856619 \dots$