Simplify ( square root of 160a^6b^2+ square root of 40a^4b^2)/( square root of 20ab)

$\frac{\sqrt{160 a^{6} b^{2}} + \sqrt{40 a^{4} b^{2}}}{\sqrt{20 a b}}$

Step 1

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{160 a^{6} b^{2}}$ as ${(160 a^{6} b^{2})}^{\frac{1}{2}}$ .

$\frac{{(160 a^{6} b^{2})}^{\frac{1}{2}} + \sqrt{40 a^{4} b^{2}}}{\sqrt{20 a b}}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{40 a^{4} b^{2}}$ as ${(40 a^{4} b^{2})}^{\frac{1}{2}}$ .

$\frac{{(160 a^{6} b^{2})}^{\frac{1}{2}} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

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

Apply the product rule to $160 a^{6} b^{2}$ .

$\frac{{(160 a^{6})}^{\frac{1}{2}} {(b^{2})}^{\frac{1}{2}} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Apply the product rule to $160 a^{6}$ .

$\frac{160^{\frac{1}{2}} {(a^{6})}^{\frac{1}{2}} {(b^{2})}^{\frac{1}{2}} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Multiply the exponents in ${(a^{6})}^{\frac{1}{2}}$ .

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

$\frac{160^{\frac{1}{2}} a^{6 (\frac{1}{2})} {(b^{2})}^{\frac{1}{2}} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Cancel the common factor of $2$ .

Factor $2$ out of $6$ .

$\frac{160^{\frac{1}{2}} a^{2 (3) \frac{1}{2}} {(b^{2})}^{\frac{1}{2}} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Cancel the common factor.

$\frac{160^{\frac{1}{2}} a^{2 \cdot 3 \frac{1}{2}} {(b^{2})}^{\frac{1}{2}} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Rewrite the expression.

$\frac{160^{\frac{1}{2}} a^{3} {(b^{2})}^{\frac{1}{2}} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Multiply the exponents in ${(b^{2})}^{\frac{1}{2}}$ .

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

$\frac{160^{\frac{1}{2}} a^{3} b^{2 (\frac{1}{2})} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{160^{\frac{1}{2}} a^{3} b^{2 (\frac{1}{2})} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Rewrite the expression.

$\frac{160^{\frac{1}{2}} a^{3} b^{1} + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Simplify.

$\frac{160^{\frac{1}{2}} a^{3} b + {(40 a^{4} b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

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

Apply the product rule to $40 a^{4} b^{2}$ .

$\frac{160^{\frac{1}{2}} a^{3} b + {(40 a^{4})}^{\frac{1}{2}} {(b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Apply the product rule to $40 a^{4}$ .

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} {(a^{4})}^{\frac{1}{2}} {(b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Multiply the exponents in ${(a^{4})}^{\frac{1}{2}}$ .

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

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{4 (\frac{1}{2})} {(b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Cancel the common factor of $2$ .

Factor $2$ out of $4$ .

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{2 (2) \frac{1}{2}} {(b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Cancel the common factor.

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{2 \cdot 2 \frac{1}{2}} {(b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Rewrite the expression.

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{2} {(b^{2})}^{\frac{1}{2}}}{\sqrt{20 a b}}$

Multiply the exponents in ${(b^{2})}^{\frac{1}{2}}$ .

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

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{2} b^{2 (\frac{1}{2})}}{\sqrt{20 a b}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{2} b^{2 (\frac{1}{2})}}{\sqrt{20 a b}}$

Rewrite the expression.

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{2} b^{1}}{\sqrt{20 a b}}$

Simplify.

$\frac{160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{2} b}{\sqrt{20 a b}}$

Factor $a^{2} b$ out of $160^{\frac{1}{2}} a^{3} b + 40^{\frac{1}{2}} a^{2} b$ .

Factor $a^{2} b$ out of $160^{\frac{1}{2}} a^{3} b$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a) + 40^{\frac{1}{2}} a^{2} b}{\sqrt{20 a b}}$

Factor $a^{2} b$ out of $40^{\frac{1}{2}} a^{2} b$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a) + a^{2} b (40^{\frac{1}{2}})}{\sqrt{20 a b}}$

Factor $a^{2} b$ out of $a^{2} b (160^{\frac{1}{2}} a) + a^{2} b (40^{\frac{1}{2}})$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{\sqrt{20 a b}}$

Step 2

Simplify the denominator.

Rewrite $20 a b$ as $2^{2} \cdot (5 a b)$ .

Factor $4$ out of $20$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{\sqrt{4 (5) a b}}$

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

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{\sqrt{2^{2} \cdot 5 a b}}$

Add parentheses.

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{\sqrt{2^{2} \cdot 5 (a b)}}$

Add parentheses.

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{\sqrt{2^{2} \cdot (5 a b)}}$

Pull terms out from under the radical.

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{2 \sqrt{5 a b}}$

Step 3

Multiply $\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{2 \sqrt{5 a b}}$ by $\frac{\sqrt{5 a b}}{\sqrt{5 a b}}$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{2 \sqrt{5 a b}} \cdot \frac{\sqrt{5 a b}}{\sqrt{5 a b}}$

Step 4

Combine and simplify the denominator.

Multiply $\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})}{2 \sqrt{5 a b}}$ by $\frac{\sqrt{5 a b}}{\sqrt{5 a b}}$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 \sqrt{5 a b} \sqrt{5 a b}}$

Move $\sqrt{5 a b}$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 (\sqrt{5 a b} \sqrt{5 a b})}$

Raise $\sqrt{5 a b}$ to the power of $1$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 ({\sqrt{5 a b}}^{1} \sqrt{5 a b})}$

Raise $\sqrt{5 a b}$ to the power of $1$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 ({\sqrt{5 a b}}^{1} {\sqrt{5 a b}}^{1})}$

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

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 {\sqrt{5 a b}}^{1 + 1}}$

Add $1$ and $1$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 {\sqrt{5 a b}}^{2}}$

Rewrite ${\sqrt{5 a b}}^{2}$ as $5 a b$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 a b}$ as ${(5 a b)}^{\frac{1}{2}}$ .

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 {({(5 a b)}^{\frac{1}{2}})}^{2}}$

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

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 {(5 a b)}^{\frac{1}{2} \cdot 2}}$

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

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 {(5 a b)}^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 {(5 a b)}^{\frac{2}{2}}}$

Rewrite the expression.

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 {(5 a b)}^{1}}$

Simplify.

$\frac{a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 (5 a b)}$

Step 5

Factor $a$ out of $a^{2} b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}$ .

$\frac{a ((a b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})) \sqrt{5 a b})}{2 (5 a b)}$

Step 6

Cancel the common factors.

Factor $a$ out of $2 (5 a b)$ .

$\frac{a ((a b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})) \sqrt{5 a b})}{a (2 (5 b))}$

Cancel the common factor.

$\frac{a ((a b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})) \sqrt{5 a b})}{a (2 (5 b))}$

Rewrite the expression.

$\frac{(a b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})) \sqrt{5 a b}}{2 (5 b)}$

Step 7

Reduce the expression by cancelling the common factors.

Cancel the common factor.

$\frac{a b (160^{\frac{1}{2}} a + 40^{\frac{1}{2}}) \sqrt{5 a b}}{2 (5 b)}$

Rewrite the expression.

$\frac{(a (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})) \sqrt{5 a b}}{2 \cdot (5)}$

Step 8

Multiply $2$ by $5$ .

$\frac{(a (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})) \sqrt{5 a b}}{10}$

Step 9

Reorder factors in $\frac{(a (160^{\frac{1}{2}} a + 40^{\frac{1}{2}})) \sqrt{5 a b}}{10}$ .

$\frac{a \sqrt{5 a b} (a \cdot 160^{\frac{1}{2}} + 40^{\frac{1}{2}})}{10}$

Simplify ( square root of 160a^6b^2+ square root of 40a^4b^2)/( square root of 20ab)

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