Basic Math Examples

$10 \sqrt[5]{7 m} \cdot (2 \sqrt{2 m})$

Step 1

Multiply $2$ by $10$ .

$20 \sqrt[5]{7 m} \sqrt{2 m}$

Step 2

Rewrite the expression using the least common index of $10$ .

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Step 2.1

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

$20 ({(2 m)}^{\frac{1}{2}} \sqrt[5]{7 m})$

Step 2.2

Rewrite ${(2 m)}^{\frac{1}{2}}$ as ${(2 m)}^{\frac{5}{10}}$ .

$20 ({(2 m)}^{\frac{5}{10}} \sqrt[5]{7 m})$

Step 2.3

Rewrite ${(2 m)}^{\frac{5}{10}}$ as $\sqrt[10]{{(2 m)}^{5}}$ .

$20 (\sqrt[10]{{(2 m)}^{5}} \sqrt[5]{7 m})$

Step 2.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{7 m}$ as ${(7 m)}^{\frac{1}{5}}$ .

$20 (\sqrt[10]{{(2 m)}^{5}} {(7 m)}^{\frac{1}{5}})$

Step 2.5

Rewrite ${(7 m)}^{\frac{1}{5}}$ as ${(7 m)}^{\frac{2}{10}}$ .

$20 (\sqrt[10]{{(2 m)}^{5}} {(7 m)}^{\frac{2}{10}})$

Step 2.6

Rewrite ${(7 m)}^{\frac{2}{10}}$ as $\sqrt[10]{{(7 m)}^{2}}$ .

$20 (\sqrt[10]{{(2 m)}^{5}} \sqrt[10]{{(7 m)}^{2}})$

Step 3

Combine using the product rule for radicals.

$20 \sqrt[10]{{(2 m)}^{5} {(7 m)}^{2}}$

Step 4

Apply the product rule to $2 m$ .

$20 \sqrt[10]{2^{5} m^{5} {(7 m)}^{2}}$

Step 5

Apply the product rule to $7 m$ .

$20 \sqrt[10]{2^{5} m^{5} \cdot 7^{2} m^{2}}$

Step 6

Multiply $m^{5}$ by $m^{2}$ by adding the exponents.

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Step 6.1

Move $m^{2}$ .

$20 \sqrt[10]{2^{5} (m^{2} m^{5}) \cdot 7^{2}}$

Step 6.2

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

$20 \sqrt[10]{2^{5} m^{2 + 5} \cdot 7^{2}}$

Step 6.3

Add $2$ and $5$ .

$20 \sqrt[10]{2^{5} m^{7} \cdot 7^{2}}$

Step 7

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

$20 \sqrt[10]{32 m^{7} \cdot 7^{2}}$

Step 8

Raise $7$ to the power of $2$ .

$20 \sqrt[10]{32 m^{7} \cdot 49}$

Step 9

Multiply $49$ by $32$ .

$20 \sqrt[10]{1568 m^{7}}$