Simplify square root of 5u^2v^7 square root of 20u^3v

$\sqrt{5 u^{2} v^{7}} \sqrt{20 u^{3} v}$

Step 1

Rewrite

$5 u^{2} v^{7}$ as

${(u v^{3})}^{2} \cdot (5 v)$ .

Factor out

$v^{6}$ .

$\sqrt{5 u^{2} (v^{6} v)} \sqrt{20 u^{3} v}$

Rewrite

$v^{6}$ as

${(v^{3})}^{2}$ .

$\sqrt{5 u^{2} ({(v^{3})}^{2} v)} \sqrt{20 u^{3} v}$

Move

$5$ .

$\sqrt{u^{2} {(v^{3})}^{2} \cdot 5 v} \sqrt{20 u^{3} v}$

Rewrite

$u^{2} {(v^{3})}^{2}$ as

${(u v^{3})}^{2}$ .

$\sqrt{{(u v^{3})}^{2} \cdot 5 v} \sqrt{20 u^{3} v}$

Add parentheses.

$\sqrt{{(u v^{3})}^{2} \cdot (5 v)} \sqrt{20 u^{3} v}$

Step 2

Pull terms out from under the radical.

$u v^{3} \sqrt{5 v} \sqrt{20 u^{3} v}$

Step 3

Rewrite

$20 u^{3} v$ as

${(2 u)}^{2} \cdot (5 u v)$ .

Factor

$4$ out of

$20$ .

$u v^{3} \sqrt{5 v} \sqrt{4 (5) u^{3} v}$

Rewrite

$4$ as

$2^{2}$ .

$u v^{3} \sqrt{5 v} \sqrt{2^{2} \cdot 5 u^{3} v}$

Factor out

$u^{2}$ .

$u v^{3} \sqrt{5 v} \sqrt{2^{2} \cdot 5 (u^{2} u) v}$

Move

$5$ .

$u v^{3} \sqrt{5 v} \sqrt{2^{2} u^{2} \cdot 5 u v}$

Rewrite

$2^{2} u^{2}$ as

${(2 u)}^{2}$ .

$u v^{3} \sqrt{5 v} \sqrt{{(2 u)}^{2} \cdot 5 u v}$

Add parentheses.

$u v^{3} \sqrt{5 v} \sqrt{{(2 u)}^{2} \cdot 5 (u v)}$

Add parentheses.

$u v^{3} \sqrt{5 v} \sqrt{{(2 u)}^{2} \cdot (5 u v)}$

Step 4

Pull terms out from under the radical.

$u v^{3} \sqrt{5 v} (2 u \sqrt{5 u v})$

Step 5

Rewrite using the commutative property of multiplication.

$2 u v^{3} \sqrt{5 v} (u \sqrt{5 u v})$

Step 6

Multiply

$u$ by

$u$ by adding the exponents.

Move

$u$ .

$2 (u \cdot u) v^{3} \sqrt{5 v} \sqrt{5 u v}$

Multiply

$u$ by

$u$ .

$2 u^{2} v^{3} \sqrt{5 v} \sqrt{5 u v}$

Step 7

Multiply

$2 u^{2} v^{3} \sqrt{5 v} \sqrt{5 u v}$ .

Combine using the product rule for radicals.

$2 u^{2} v^{3} \sqrt{5 u v (5 v)}$

Multiply

$5$ by

$5$ .

$2 u^{2} v^{3} \sqrt{25 u v \cdot v}$

Raise

$v$ to the power of

$1$ .

$2 u^{2} v^{3} \sqrt{25 u (v^{1} v)}$

Raise

$v$ to the power of

$1$ .

$2 u^{2} v^{3} \sqrt{25 u (v^{1} v^{1})}$

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 u^{2} v^{3} \sqrt{25 u v^{1 + 1}}$

Add

$1$ and

$1$ .

$2 u^{2} v^{3} \sqrt{25 u v^{2}}$

Step 8

Rewrite

$25 u v^{2}$ as

${(5 v)}^{2} u$ .

Rewrite

$25$ as

$5^{2}$ .

$2 u^{2} v^{3} \sqrt{5^{2} u v^{2}}$

Move

$u$ .

$2 u^{2} v^{3} \sqrt{5^{2} v^{2} u}$

Rewrite

$5^{2} v^{2}$ as

${(5 v)}^{2}$ .

$2 u^{2} v^{3} \sqrt{{(5 v)}^{2} u}$

Step 9

Pull terms out from under the radical.

$2 u^{2} v^{3} (5 v \sqrt{u})$

Step 10

Multiply

$v^{3}$ by

$v$ by adding the exponents.

Move

$v$ .

$2 u^{2} (v \cdot v^{3}) (5 \sqrt{u})$

Multiply

$v$ by

$v^{3}$ .

Raise

$v$ to the power of

$1$ .

$2 u^{2} (v^{1} v^{3}) (5 \sqrt{u})$

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 u^{2} v^{1 + 3} (5 \sqrt{u})$

Add

$1$ and

$3$ .

$2 u^{2} v^{4} (5 \sqrt{u})$

Step 11

Multiply

$5$ by

$2$ .

$10 u^{2} v^{4} \sqrt{u}$

Simplify square root of 5u^2v^7 square root of 20u^3v

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