Simplify (8 square root of 6mn+6 square root of 8mn)/(2 square root of 2mn)

$\frac{8 \sqrt{6 m n} + 6 \sqrt{8 m n}}{2 \sqrt{2 m n}}$

Step 1

Cancel the common factor of $8 \sqrt{6 m n} + 6 \sqrt{8 m n}$ and $2$ .

Factor $2$ out of $8 \sqrt{6 m n}$ .

$\frac{2 (4 \sqrt{6 m n}) + 6 \sqrt{8 m n}}{2 \sqrt{2 m n}}$

Factor $2$ out of $6 \sqrt{8 m n}$ .

$\frac{2 (4 \sqrt{6 m n}) + 2 (3 \sqrt{8 m n})}{2 \sqrt{2 m n}}$

Factor $2$ out of $2 (4 \sqrt{6 m n}) + 2 (3 \sqrt{8 m n})$ .

$\frac{2 (4 \sqrt{6 m n} + 3 \sqrt{8 m n})}{2 \sqrt{2 m n}}$

Cancel the common factors.

Factor $2$ out of $2 \sqrt{2 m n}$ .

$\frac{2 (4 \sqrt{6 m n} + 3 \sqrt{8 m n})}{2 (\sqrt{2 m n})}$

Cancel the common factor.

$\frac{2 (4 \sqrt{6 m n} + 3 \sqrt{8 m n})}{2 \sqrt{2 m n}}$

Rewrite the expression.

$\frac{4 \sqrt{6 m n} + 3 \sqrt{8 m n}}{\sqrt{2 m n}}$

Step 2

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6 m n}$ as ${(6 m n)}^{\frac{1}{2}}$ .

$\frac{4 {(6 m n)}^{\frac{1}{2}} + 3 \sqrt{8 m n}}{\sqrt{2 m n}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 m n}$ as ${(8 m n)}^{\frac{1}{2}}$ .

$\frac{4 {(6 m n)}^{\frac{1}{2}} + 3 {(8 m n)}^{\frac{1}{2}}}{\sqrt{2 m n}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $6 m n$ .

$\frac{4 ({(6 m)}^{\frac{1}{2}} n^{\frac{1}{2}}) + 3 {(8 m n)}^{\frac{1}{2}}}{\sqrt{2 m n}}$

Apply the product rule to $6 m$ .

$\frac{4 (6^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}}) + 3 {(8 m n)}^{\frac{1}{2}}}{\sqrt{2 m n}}$

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $8 m n$ .

$\frac{4 \cdot 6^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}} + 3 ({(8 m)}^{\frac{1}{2}} n^{\frac{1}{2}})}{\sqrt{2 m n}}$

Apply the product rule to $8 m$ .

$\frac{4 \cdot 6^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}} + 3 (8^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}})}{\sqrt{2 m n}}$

Factor $m^{\frac{1}{2}} n^{\frac{1}{2}}$ out of $4 \cdot 6^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}}$ .

Factor $m^{\frac{1}{2}} n^{\frac{1}{2}}$ out of $4 \cdot 6^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}}$ .

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}}) + 3 \cdot 8^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}}}{\sqrt{2 m n}}$

Factor $m^{\frac{1}{2}} n^{\frac{1}{2}}$ out of $3 \cdot 8^{\frac{1}{2}} m^{\frac{1}{2}} n^{\frac{1}{2}}$ .

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}}) + m^{\frac{1}{2}} n^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}})}{\sqrt{2 m n}}$

Factor $m^{\frac{1}{2}} n^{\frac{1}{2}}$ out of $m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}}) + m^{\frac{1}{2}} n^{\frac{1}{2}} (3 \cdot 8^{\frac{1}{2}})$ .

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}})}{\sqrt{2 m n}}$

Step 3

Multiply $\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}})}{\sqrt{2 m n}}$ by $\frac{\sqrt{2 m n}}{\sqrt{2 m n}}$ .

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}})}{\sqrt{2 m n}} \cdot \frac{\sqrt{2 m n}}{\sqrt{2 m n}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}})}{\sqrt{2 m n}}$ by $\frac{\sqrt{2 m n}}{\sqrt{2 m n}}$ .

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{\sqrt{2 m n} \sqrt{2 m n}}$

Raise $\sqrt{2 m n}$ to the power of $1$ .

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{\sqrt{2 m n}}^{1} \sqrt{2 m n}}$

Raise $\sqrt{2 m n}$ to the power of $1$ .

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{\sqrt{2 m n}}^{1} {\sqrt{2 m n}}^{1}}$

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

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{\sqrt{2 m n}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{\sqrt{2 m n}}^{2}}$

Rewrite ${\sqrt{2 m n}}^{2}$ as $2 m n$ .

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

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{({(2 m n)}^{\frac{1}{2}})}^{2}}$

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

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{(2 m n)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{(2 m n)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{(2 m n)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{{(2 m n)}^{1}}$

Simplify.

$\frac{m^{\frac{1}{2}} n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m n}$

Step 5

Simplify.

Move $m^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{n^{\frac{1}{2}} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m n m^{- \frac{1}{2}}}$

Move $n^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m n m^{- \frac{1}{2}} n^{- \frac{1}{2}}}$

Step 6

Simplify the denominator.

Multiply $m$ by $m^{- \frac{1}{2}}$ by adding the exponents.

Move $m^{- \frac{1}{2}}$ .

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 (m^{- \frac{1}{2}} m) n \cdot n^{- \frac{1}{2}}}$

Multiply $m^{- \frac{1}{2}}$ by $m$ .

Raise $m$ to the power of $1$ .

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 (m^{- \frac{1}{2}} m^{1}) n \cdot n^{- \frac{1}{2}}}$

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

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{- \frac{1}{2} + 1} n \cdot n^{- \frac{1}{2}}}$

Write $1$ as a fraction with a common denominator.

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{- \frac{1}{2} + \frac{2}{2}} n \cdot n^{- \frac{1}{2}}}$

Combine the numerators over the common denominator.

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{- 1 + 2}{2}} n \cdot n^{- \frac{1}{2}}}$

Add $- 1$ and $2$ .

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{1}{2}} n \cdot n^{- \frac{1}{2}}}$

Multiply $n$ by $n^{- \frac{1}{2}}$ by adding the exponents.

Move $n^{- \frac{1}{2}}$ .

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{1}{2}} (n^{- \frac{1}{2}} n)}$

Multiply $n^{- \frac{1}{2}}$ by $n$ .

Raise $n$ to the power of $1$ .

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{1}{2}} (n^{- \frac{1}{2}} n^{1})}$

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

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{1}{2}} n^{- \frac{1}{2} + 1}}$

Write $1$ as a fraction with a common denominator.

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{1}{2}} n^{- \frac{1}{2} + \frac{2}{2}}}$

Combine the numerators over the common denominator.

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{1}{2}} n^{\frac{- 1 + 2}{2}}}$

Add $- 1$ and $2$ .

$\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{1}{2}} n^{\frac{1}{2}}}$

Step 7

Reorder factors in $\frac{(4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}}) \sqrt{2 m n}}{2 m^{\frac{1}{2}} n^{\frac{1}{2}}}$ .

$\frac{\sqrt{2 m n} (4 \cdot 6^{\frac{1}{2}} + 3 \cdot 8^{\frac{1}{2}})}{2 m^{\frac{1}{2}} n^{\frac{1}{2}}}$

Simplify (8 square root of 6mn+6 square root of 8mn)/(2 square root of 2mn)

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