Algebra Examples

$R = \frac{\sqrt[6]{25} \cdot \sqrt[4]{25} \cdot \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite the expression using the least common index of $12$ .

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Step 1.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{25}$ as $25^{\frac{1}{6}}$ .

$R = \frac{25^{\frac{1}{6}} \cdot \sqrt[4]{25} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.1.2

Rewrite $25^{\frac{1}{6}}$ as $25^{\frac{2}{12}}$ .

$R = \frac{25^{\frac{2}{12}} \cdot \sqrt[4]{25} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.1.3

Rewrite $25^{\frac{2}{12}}$ as $\sqrt[12]{25^{2}}$ .

$R = \frac{\sqrt[12]{25^{2}} \cdot \sqrt[4]{25} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.1.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{25}$ as $25^{\frac{1}{4}}$ .

$R = \frac{\sqrt[12]{25^{2}} \cdot 25^{\frac{1}{4}} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.1.5

Rewrite $25^{\frac{1}{4}}$ as $25^{\frac{3}{12}}$ .

$R = \frac{\sqrt[12]{25^{2}} \cdot 25^{\frac{3}{12}} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.1.6

Rewrite $25^{\frac{3}{12}}$ as $\sqrt[12]{25^{3}}$ .

$R = \frac{\sqrt[12]{25^{2}} \cdot \sqrt[12]{25^{3}} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.2

Combine using the product rule for radicals.

$R = \frac{\sqrt[12]{25^{2} \cdot 25^{3}} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.3

Multiply $25^{2}$ by $25^{3}$ by adding the exponents.

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Step 1.3.1

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

$R = \frac{\sqrt[12]{25^{2 + 3}} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.3.2

Add $2$ and $3$ .

$R = \frac{\sqrt[12]{25^{5}} \sqrt[3]{25}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.4

Rewrite the expression using the least common index of $12$ .

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Step 1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{25}$ as $25^{\frac{1}{3}}$ .

$R = \frac{\sqrt[12]{25^{5}} \cdot 25^{\frac{1}{3}}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.4.2

Rewrite $25^{\frac{1}{3}}$ as $25^{\frac{4}{12}}$ .

$R = \frac{\sqrt[12]{25^{5}} \cdot 25^{\frac{4}{12}}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.4.3

Rewrite $25^{\frac{4}{12}}$ as $\sqrt[12]{25^{4}}$ .

$R = \frac{\sqrt[12]{25^{5}} \sqrt[12]{25^{4}}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.5

Combine using the product rule for radicals.

$R = \frac{\sqrt[12]{25^{5} \cdot 25^{4}}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.6

Multiply $25^{5}$ by $25^{4}$ by adding the exponents.

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Step 1.6.1

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

$R = \frac{\sqrt[12]{25^{5 + 4}}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 1.6.2

Add $5$ and $4$ .

$R = \frac{\sqrt[12]{25^{9}}}{5 \cdot \sqrt[5]{25} \cdot \sqrt[20]{25}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the least common index of $20$ .

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Step 2.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{25}$ as $25^{\frac{1}{5}}$ .

$R = \frac{\sqrt[12]{25^{9}}}{5 \cdot (\sqrt[20]{25} \cdot 25^{\frac{1}{5}})}$

Step 2.1.2

Rewrite $25^{\frac{1}{5}}$ as $25^{\frac{4}{20}}$ .

$R = \frac{\sqrt[12]{25^{9}}}{5 \cdot (\sqrt[20]{25} \cdot 25^{\frac{4}{20}})}$

Step 2.1.3

Rewrite $25^{\frac{4}{20}}$ as $\sqrt[20]{25^{4}}$ .

$R = \frac{\sqrt[12]{25^{9}}}{5 \cdot (\sqrt[20]{25} \sqrt[20]{25^{4}})}$

Step 2.2

Combine using the product rule for radicals.

$R = \frac{\sqrt[12]{25^{9}}}{5 \cdot \sqrt[20]{25 \cdot 25^{4}}}$

Step 2.3

Multiply $25$ by $25^{4}$ by adding the exponents.

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Step 2.3.1

Multiply $25$ by $25^{4}$ .

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Step 2.3.1.1

Raise $25$ to the power of $1$ .

$R = \frac{\sqrt[12]{25^{9}}}{5 \cdot \sqrt[20]{25^{1} \cdot 25^{4}}}$

Step 2.3.1.2

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

$R = \frac{\sqrt[12]{25^{9}}}{5 \cdot \sqrt[20]{25^{1 + 4}}}$

Step 2.3.2

Add $1$ and $4$ .

$R = \frac{\sqrt[12]{25^{9}}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3

Simplify the numerator.

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Step 3.1

Raise $25$ to the power of $9$ .

$R = \frac{\sqrt[12]{3814697265625}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.2

Rewrite $3814697265625$ as $5^{12} \cdot 15625$ .

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Step 3.2.1

Factor $244140625$ out of $3814697265625$ .

$R = \frac{\sqrt[12]{244140625 (15625)}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.2.2

Rewrite $244140625$ as $5^{12}$ .

$R = \frac{\sqrt[12]{5^{12} \cdot 15625}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.3

Pull terms out from under the radical.

$R = \frac{5 \sqrt[12]{15625}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.4

Rewrite $15625$ as $25^{3}$ .

$R = \frac{5 \sqrt[12]{25^{3}}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.5

Rewrite $\sqrt[12]{25^{3}}$ as $\sqrt[4]{\sqrt[3]{25^{3}}}$ .

$R = \frac{5 \sqrt[4]{\sqrt[3]{25^{3}}}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.6

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

$R = \frac{5 \sqrt[4]{25}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.7

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

$R = \frac{5 \sqrt[4]{5^{2}}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.8

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

$R = \frac{5 \sqrt{\sqrt{5^{2}}}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 3.9

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

$R = \frac{5 \sqrt{5}}{5 \cdot \sqrt[20]{25^{5}}}$

Step 4

Simplify the denominator.

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Step 4.1

Raise $25$ to the power of $5$ .

$R = \frac{5 \sqrt{5}}{5 \sqrt[20]{9765625}}$

Step 4.2

Rewrite $9765625$ as $25^{5}$ .

$R = \frac{5 \sqrt{5}}{5 \sqrt[20]{25^{5}}}$

Step 4.3

Rewrite $\sqrt[20]{25^{5}}$ as $\sqrt[4]{\sqrt[5]{25^{5}}}$ .

$R = \frac{5 \sqrt{5}}{5 \sqrt[4]{\sqrt[5]{25^{5}}}}$

Step 4.4

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

$R = \frac{5 \sqrt{5}}{5 \sqrt[4]{25}}$

Step 4.5

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

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

Step 4.6

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

$R = \frac{5 \sqrt{5}}{5 \sqrt{\sqrt{5^{2}}}}$

Step 4.7

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

$R = \frac{5 \sqrt{5}}{5 \sqrt{5}}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of $5$ .

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Step 5.1.1

Cancel the common factor.

$R = \frac{5 \sqrt{5}}{5 \sqrt{5}}$

Step 5.1.2

Rewrite the expression.

$R = \frac{\sqrt{5}}{\sqrt{5}}$

Step 5.2

Cancel the common factor of $\sqrt{5}$ .

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Step 5.2.1

Cancel the common factor.

$R = \frac{\sqrt{5}}{\sqrt{5}}$

Step 5.2.2

Rewrite the expression.

$R = 1$