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Simplify square root of 5a^4 square root of 20a^3

$\sqrt{5 a^{4}} \sqrt{20 a^{3}}$

Step 1

Rewrite

$5 a^{4}$ as

${(a^{2})}^{2} \cdot 5$ .

Rewrite

$a^{4}$ as

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

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

Reorder

$5$ and

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

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

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

Step 2

Pull terms out from under the radical.

$a^{2} \sqrt{5} \sqrt{20 a^{3}}$

Step 3

Rewrite

$20 a^{3}$ as

${(2 a)}^{2} \cdot (5 a)$ .

Factor

$4$ out of

$20$ .

$a^{2} \sqrt{5} \sqrt{4 (5) a^{3}}$

Rewrite

$4$ as

$2^{2}$ .

$a^{2} \sqrt{5} \sqrt{2^{2} \cdot 5 a^{3}}$

Factor out

$a^{2}$ .

$a^{2} \sqrt{5} \sqrt{2^{2} \cdot 5 (a^{2} a)}$

Move

$5$ .

$a^{2} \sqrt{5} \sqrt{2^{2} a^{2} \cdot 5 a}$

Rewrite

$2^{2} a^{2}$ as

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

$a^{2} \sqrt{5} \sqrt{{(2 a)}^{2} \cdot 5 a}$

Add parentheses.

$a^{2} \sqrt{5} \sqrt{{(2 a)}^{2} \cdot (5 a)}$

$a^{2} \sqrt{5} \sqrt{{(2 a)}^{2} \cdot (5 a)}$

Step 4

Pull terms out from under the radical.

$a^{2} \sqrt{5} (2 a \sqrt{5 a})$

Step 5

Multiply

$a^{2}$ by

$a$ by adding the exponents.

Move

$a$ .

$a \cdot a^{2} \sqrt{5} (2 \sqrt{5 a})$

Multiply

$a$ by

$a^{2}$ .

Raise

$a$ to the power of

$1$ .

$a^{1} a^{2} \sqrt{5} (2 \sqrt{5 a})$

Use the power rule

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

$a^{1 + 2} \sqrt{5} (2 \sqrt{5 a})$

$a^{1 + 2} \sqrt{5} (2 \sqrt{5 a})$

Add

$1$ and

$2$ .

$a^{3} \sqrt{5} (2 \sqrt{5 a})$

$a^{3} \sqrt{5} (2 \sqrt{5 a})$

Step 6

Multiply

$a^{3} \sqrt{5} (2 \sqrt{5 a})$ .

Combine using the product rule for radicals.

$a^{3} (2 \sqrt{5 (5 a)})$

Multiply

$5$ by

$5$ .

$a^{3} (2 \sqrt{25 a})$

$a^{3} (2 \sqrt{25 a})$

Step 7

Simplify the expression.

Rewrite using the commutative property of multiplication.

$2 a^{3} \sqrt{25 a}$

Rewrite

$25$ as

$5^{2}$ .

$2 a^{3} \sqrt{5^{2} a}$

$2 a^{3} \sqrt{5^{2} a}$

Step 8

Pull terms out from under the radical.

$2 a^{3} (5 \sqrt{a})$

Step 9

Multiply

$5$ by

$2$ .

$10 a^{3} \sqrt{a}$

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