Basic Math Examples

$4 \sqrt{7 m} \cdot (5 \sqrt{\frac{1}{7}} m^{5})$

Step 1

Rewrite $\sqrt{\frac{1}{7}}$ as $\frac{\sqrt{1}}{\sqrt{7}}$ .

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{1}}{\sqrt{7}} m^{5})$

Step 2

Any root of $1$ is $1$ .

$4 \sqrt{7 m} \cdot (5 \frac{1}{\sqrt{7}} m^{5})$

Step 3

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$4 \sqrt{7 m} \cdot (5 (\frac{1}{\sqrt{7}} \cdot \frac{\sqrt{7}}{\sqrt{7}}) m^{5})$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{1}{\sqrt{7}}$ by $\frac{\sqrt{7}}{\sqrt{7}}$ .

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{\sqrt{7} \sqrt{7}} m^{5})$

Step 4.2

Raise $\sqrt{7}$ to the power of $1$ .

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{{\sqrt{7}}^{1} \sqrt{7}} m^{5})$

Step 4.3

Raise $\sqrt{7}$ to the power of $1$ .

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{{\sqrt{7}}^{1} {\sqrt{7}}^{1}} m^{5})$

Step 4.4

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

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{{\sqrt{7}}^{1 + 1}} m^{5})$

Step 4.5

Add $1$ and $1$ .

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{{\sqrt{7}}^{2}} m^{5})$

Step 4.6

Rewrite ${\sqrt{7}}^{2}$ as $7$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{{(7^{\frac{1}{2}})}^{2}} m^{5})$

Step 4.6.2

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

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{7^{\frac{1}{2} \cdot 2}} m^{5})$

Step 4.6.3

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

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{7^{\frac{2}{2}}} m^{5})$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{7^{\frac{2}{2}}} m^{5})$

Step 4.6.4.2

Rewrite the expression.

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{7^{1}} m^{5})$

Step 4.6.5

Evaluate the exponent.

$4 \sqrt{7 m} \cdot (5 \frac{\sqrt{7}}{7} m^{5})$

Step 5

Combine $5$ and $\frac{\sqrt{7}}{7}$ .

$4 \sqrt{7 m} \cdot (\frac{5 \sqrt{7}}{7} m^{5})$

Step 6

Combine $\frac{5 \sqrt{7}}{7}$ and $m^{5}$ .

$4 \sqrt{7 m} \cdot \frac{5 \sqrt{7} m^{5}}{7}$

Step 7

Multiply $4 \sqrt{7 m} \frac{5 \sqrt{7} m^{5}}{7}$ .

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Step 7.1

Combine $\frac{5 \sqrt{7} m^{5}}{7}$ and $4$ .

$\frac{5 \sqrt{7} m^{5} \cdot 4}{7} \sqrt{7 m}$

Step 7.2

Multiply $4$ by $5$ .

$\frac{20 \sqrt{7} m^{5}}{7} \sqrt{7 m}$

Step 7.3

Combine $\frac{20 \sqrt{7} m^{5}}{7}$ and $\sqrt{7 m}$ .

$\frac{20 \sqrt{7} m^{5} \sqrt{7 m}}{7}$

Step 7.4

Combine using the product rule for radicals.

$\frac{20 m^{5} \sqrt{7 (7 m)}}{7}$

Step 7.5

Multiply $7$ by $7$ .

$\frac{20 m^{5} \sqrt{49 m}}{7}$

Step 8

Simplify the numerator.

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Step 8.1

Rewrite $49$ as $7^{2}$ .

$\frac{20 m^{5} \sqrt{7^{2} m}}{7}$

Step 8.2

Pull terms out from under the radical.

$\frac{20 m^{5} \cdot 7 \sqrt{m}}{7}$

Step 8.3

Multiply $7$ by $20$ .

$\frac{140 m^{5} \sqrt{m}}{7}$

Step 9

Cancel the common factor of $140$ and $7$ .

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Step 9.1

Factor $7$ out of $140 m^{5} \sqrt{m}$ .

$\frac{7 (20 m^{5} \sqrt{m})}{7}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $7$ out of $7$ .

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

Step 9.2.2

Cancel the common factor.

$\frac{7 (20 m^{5} \sqrt{m})}{7 \cdot 1}$

Step 9.2.3

Rewrite the expression.

$\frac{20 m^{5} \sqrt{m}}{1}$

Step 9.2.4

Divide $20 m^{5} \sqrt{m}$ by $1$ .

$20 m^{5} \sqrt{m}$