Algebra Examples

$\sqrt[5]{\frac{5 m^{7}}{16 m^{2}}}$

Step 1

Reduce the expression $\frac{5 m^{7}}{16 m^{2}}$ by cancelling the common factors.

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Step 1.1

Factor $m^{2}$ out of $5 m^{7}$ .

$\sqrt[5]{\frac{m^{2} (5 m^{5})}{16 m^{2}}}$

Step 1.2

Factor $m^{2}$ out of $16 m^{2}$ .

$\sqrt[5]{\frac{m^{2} (5 m^{5})}{m^{2} \cdot 16}}$

Step 1.3

Cancel the common factor.

$\sqrt[5]{\frac{m^{2} (5 m^{5})}{m^{2} \cdot 16}}$

Step 1.4

Rewrite the expression.

$\sqrt[5]{\frac{5 m^{5}}{16}}$

Step 2

Rewrite $\sqrt[5]{\frac{5 m^{5}}{16}}$ as $\frac{\sqrt[5]{5 m^{5}}}{\sqrt[5]{16}}$ .

$\frac{\sqrt[5]{5 m^{5}}}{\sqrt[5]{16}}$

Step 3

Simplify the numerator.

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Step 3.1

Reorder $5$ and $m^{5}$ .

$\frac{\sqrt[5]{m^{5} \cdot 5}}{\sqrt[5]{16}}$

Step 3.2

Pull terms out from under the radical.

$\frac{m \sqrt[5]{5}}{\sqrt[5]{16}}$

Step 4

Multiply $\frac{m \sqrt[5]{5}}{\sqrt[5]{16}}$ by $\frac{{\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{4}}$ .

$\frac{m \sqrt[5]{5}}{\sqrt[5]{16}} \cdot \frac{{\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{4}}$

Step 5

Combine and simplify the denominator.

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Step 5.1

Multiply $\frac{m \sqrt[5]{5}}{\sqrt[5]{16}}$ by $\frac{{\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{4}}$ .

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{\sqrt[5]{16} {\sqrt[5]{16}}^{4}}$

Step 5.2

Raise $\sqrt[5]{16}$ to the power of $1$ .

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{1} {\sqrt[5]{16}}^{4}}$

Step 5.3

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

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{1 + 4}}$

Step 5.4

Add $1$ and $4$ .

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{{\sqrt[5]{16}}^{5}}$

Step 5.5

Rewrite ${\sqrt[5]{16}}^{5}$ as $16$ .

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{16}$ as $16^{\frac{1}{5}}$ .

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{{(16^{\frac{1}{5}})}^{5}}$

Step 5.5.2

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

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{16^{\frac{1}{5} \cdot 5}}$

Step 5.5.3

Combine $\frac{1}{5}$ and $5$ .

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{16^{\frac{5}{5}}}$

Step 5.5.4

Cancel the common factor of $5$ .

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Step 5.5.4.1

Cancel the common factor.

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{16^{\frac{5}{5}}}$

Step 5.5.4.2

Rewrite the expression.

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{16^{1}}$

Step 5.5.5

Evaluate the exponent.

$\frac{m \sqrt[5]{5} {\sqrt[5]{16}}^{4}}{16}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite ${\sqrt[5]{16}}^{4}$ as $\sqrt[5]{16^{4}}$ .

$\frac{m \sqrt[5]{5} \sqrt[5]{16^{4}}}{16}$

Step 6.2

Raise $16$ to the power of $4$ .

$\frac{m \sqrt[5]{5} \sqrt[5]{65536}}{16}$

Step 6.3

Rewrite $65536$ as $8^{5} \cdot 2$ .

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Step 6.3.1

Factor $32768$ out of $65536$ .

$\frac{m \sqrt[5]{5} \sqrt[5]{32768 (2)}}{16}$

Step 6.3.2

Rewrite $32768$ as $8^{5}$ .

$\frac{m \sqrt[5]{5} \sqrt[5]{8^{5} \cdot 2}}{16}$

Step 6.4

Pull terms out from under the radical.

$\frac{m \sqrt[5]{5} \cdot 8 \sqrt[5]{2}}{16}$

Step 6.5

Combine exponents.

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Step 6.5.1

Combine using the product rule for radicals.

$\frac{m \cdot 8 \sqrt[5]{5 \cdot 2}}{16}$

Step 6.5.2

Multiply $5$ by $2$ .

$\frac{m \cdot 8 \sqrt[5]{10}}{16}$

Step 7

Cancel the common factor of $8$ and $16$ .

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Step 7.1

Factor $8$ out of $m \cdot 8 \sqrt[5]{10}$ .

$\frac{8 (m \sqrt[5]{10})}{16}$

Step 7.2

Cancel the common factors.

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Step 7.2.1

Factor $8$ out of $16$ .

$\frac{8 (m \sqrt[5]{10})}{8 \cdot 2}$

Step 7.2.2

Cancel the common factor.

$\frac{8 (m \sqrt[5]{10})}{8 \cdot 2}$

Step 7.2.3

Rewrite the expression.

$\frac{m \sqrt[5]{10}}{2}$