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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