Simplify the Radical Expression ( square root of 8m^2)/( square root of 6m)

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

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.

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

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

Factor $2$ out of $6 m$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 3

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

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

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

Cancel the common factors.

Factor $m$ out of $3 m$ .

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

Cancel the common factor.

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

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.

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

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

Pull terms out from under the radical.

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

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

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

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.

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Step 8

Combine using the product rule for radicals.

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

Simplify the Radical Expression ( square root of 8m^2)/( square root of 6m)

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