Algebra Examples

$\frac{\sqrt{8 m^{2}}}{\sqrt{6 m}}$

Step 1

Combine $\sqrt{8 m^{2}}$ and $\sqrt{6 m}$ into a single radical.

$\sqrt{\frac{8 m^{2}}{6 m}}$

Step 2

Reduce the expression $\frac{8 m^{2}}{6 m}$ by cancelling the common factors.

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Step 2.1

Factor $2$ out of $8 m^{2}$ .

$\sqrt{\frac{2 (4 m^{2})}{6 m}}$

Step 2.2

Factor $2$ out of $6 m$ .

$\sqrt{\frac{2 (4 m^{2})}{2 (3 m)}}$

Step 2.3

Cancel the common factor.

$\sqrt{\frac{2 (4 m^{2})}{2 (3 m)}}$

Step 2.4

Rewrite the expression.

$\sqrt{\frac{4 m^{2}}{3 m}}$

Step 3

Cancel the common factor of $m^{2}$ and $m$ .

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Step 3.1

Factor $m$ out of $4 m^{2}$ .

$\sqrt{\frac{m (4 m)}{3 m}}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $m$ out of $3 m$ .

$\sqrt{\frac{m (4 m)}{m \cdot 3}}$

Step 3.2.2

Cancel the common factor.

$\sqrt{\frac{m (4 m)}{m \cdot 3}}$

Step 3.2.3

Rewrite the expression.

$\sqrt{\frac{4 m}{3}}$

Step 4

Rewrite $\sqrt{\frac{4 m}{3}}$ as $\frac{\sqrt{4 m}}{\sqrt{3}}$ .

$\frac{\sqrt{4 m}}{\sqrt{3}}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2} m}}{\sqrt{3}}$

Step 5.2

Pull terms out from under the radical.

$\frac{2 \sqrt{m}}{\sqrt{3}}$

Step 6

Multiply $\frac{2 \sqrt{m}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2 \sqrt{m}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 7

Combine and simplify the denominator.

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Step 7.1

Multiply $\frac{2 \sqrt{m}}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

$\frac{2 \sqrt{m} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 7.2

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

$\frac{2 \sqrt{m} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 7.3

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

$\frac{2 \sqrt{m} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 7.4

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

$\frac{2 \sqrt{m} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$\frac{2 \sqrt{m} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 7.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 7.6.1

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

$\frac{2 \sqrt{m} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 7.6.2

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

$\frac{2 \sqrt{m} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 7.6.3

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

$\frac{2 \sqrt{m} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4

Cancel the common factor of $2$ .

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Step 7.6.4.1

Cancel the common factor.

$\frac{2 \sqrt{m} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 7.6.4.2

Rewrite the expression.

$\frac{2 \sqrt{m} \sqrt{3}}{3^{1}}$

Step 7.6.5

Evaluate the exponent.

$\frac{2 \sqrt{m} \sqrt{3}}{3}$

Step 8

Combine using the product rule for radicals.

$\frac{2 \sqrt{3 m}}{3}$