Pre-Algebra Examples

$\frac{\sqrt{75 x^{4}}}{\sqrt{\sqrt{3}}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $75 x^{4}$ as ${(5 x^{2})}^{2} \cdot 3$ .

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Step 1.1.1

Factor $25$ out of $75$ .

$\frac{\sqrt{25 (3) x^{4}}}{\sqrt{\sqrt{3}}}$

Step 1.1.2

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

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

Step 1.1.3

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

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

Step 1.1.4

Move $3$ .

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

Step 1.1.5

Rewrite $5^{2} {(x^{2})}^{2}$ as ${(5 x^{2})}^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

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

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

Step 3

Multiply $\frac{5 x^{2} \sqrt{3}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

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

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{5 x^{2} \sqrt{3}}{\sqrt[4]{3}}$ by $\frac{{\sqrt[4]{3}}^{3}}{{\sqrt[4]{3}}^{3}}$ .

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

Step 4.2

Raise $\sqrt[4]{3}$ to the power of $1$ .

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

Step 4.3

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

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

Step 4.4

Add $1$ and $3$ .

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

Step 4.5

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

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Step 4.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3}$ as $3^{\frac{1}{4}}$ .

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

Step 4.5.2

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

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

Step 4.5.3

Combine $\frac{1}{4}$ and $4$ .

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

Step 4.5.4

Cancel the common factor of $4$ .

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Step 4.5.4.1

Cancel the common factor.

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

Step 4.5.4.2

Rewrite the expression.

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

Step 4.5.5

Evaluate the exponent.

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

Raise $3$ to the power of $3$ .

$\frac{5 x^{2} \sqrt{3} \sqrt[4]{27}}{3}$

Step 5.3

Combine exponents.

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Step 5.3.1

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

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Step 5.3.1.1

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

$\frac{5 x^{2} (\sqrt[4]{27} \cdot 3^{\frac{1}{2}})}{3}$

Step 5.3.1.2

Rewrite $3^{\frac{1}{2}}$ as $3^{\frac{2}{4}}$ .

$\frac{5 x^{2} (\sqrt[4]{27} \cdot 3^{\frac{2}{4}})}{3}$

Step 5.3.1.3

Rewrite $3^{\frac{2}{4}}$ as $\sqrt[4]{3^{2}}$ .

$\frac{5 x^{2} (\sqrt[4]{27} \sqrt[4]{3^{2}})}{3}$

Step 5.3.2

Combine using the product rule for radicals.

$\frac{5 x^{2} \sqrt[4]{27 \cdot 3^{2}}}{3}$

Step 5.3.3

Rewrite $27$ as $3^{3}$ .

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

Step 5.3.4

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

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

Step 5.3.5

Add $3$ and $2$ .

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

Step 5.4

Raise $3$ to the power of $5$ .

$\frac{5 x^{2} \sqrt[4]{243}}{3}$

Step 5.5

Rewrite $243$ as $3^{4} \cdot 3$ .

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Step 5.5.1

Factor $81$ out of $243$ .

$\frac{5 x^{2} \sqrt[4]{81 (3)}}{3}$

Step 5.5.2

Rewrite $81$ as $3^{4}$ .

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

Step 5.6

Pull terms out from under the radical.

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

Step 6

Multiply $3$ by $5$ .

$\frac{15 x^{2} \sqrt[4]{3}}{3}$

Step 7

Cancel the common factor of $15$ and $3$ .

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Step 7.1

Factor $3$ out of $15 x^{2} \sqrt[4]{3}$ .

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $3$ out of $3$ .

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

Step 7.2.2

Cancel the common factor.

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

Step 7.2.3

Rewrite the expression.

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

Step 7.2.4

Divide $5 x^{2} \sqrt[4]{3}$ by $1$ .

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