Simplify 3 square root of 5a^3* square root of 50a^7

$3 \sqrt{5 a^{3}} \cdot \sqrt{50 a^{7}}$

Step 1

Rewrite $5 a^{3}$ as $a^{2} \cdot (5 a)$ .

Factor out $a^{2}$ .

$3 \sqrt{5 (a^{2} a)} \cdot \sqrt{50 a^{7}}$

Reorder $5$ and $a^{2}$ .

$3 \sqrt{a^{2} \cdot 5 a} \cdot \sqrt{50 a^{7}}$

Add parentheses.

$3 \sqrt{a^{2} \cdot (5 a)} \cdot \sqrt{50 a^{7}}$

Step 2

Pull terms out from under the radical.

$3 (a \sqrt{5 a}) \cdot \sqrt{50 a^{7}}$

Step 3

Rewrite $50 a^{7}$ as ${(5 a^{3})}^{2} \cdot (2 a)$ .

Factor $25$ out of $50$ .

$3 a \sqrt{5 a} \cdot \sqrt{25 (2) a^{7}}$

Rewrite $25$ as $5^{2}$ .

$3 a \sqrt{5 a} \cdot \sqrt{5^{2} \cdot 2 a^{7}}$

Factor out $a^{6}$ .

$3 a \sqrt{5 a} \cdot \sqrt{5^{2} \cdot 2 (a^{6} a)}$

Rewrite $a^{6}$ as ${(a^{3})}^{2}$ .

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

Move $2$ .

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

Rewrite $5^{2} {(a^{3})}^{2}$ as ${(5 a^{3})}^{2}$ .

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

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $a$ by $a^{3}$ by adding the exponents.

Move $a^{3}$ .

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

Multiply $a^{3}$ by $a$ .

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

$3 (a^{3} a^{1}) \sqrt{5 a} \cdot (5 \sqrt{2 a})$

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

$3 a^{3 + 1} \sqrt{5 a} \cdot (5 \sqrt{2 a})$

Add $3$ and $1$ .

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

Step 6

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

Multiply $5$ by $3$ .

$15 a^{4} \sqrt{5 a} \sqrt{2 a}$

Combine using the product rule for radicals.

$15 a^{4} \sqrt{2 a (5 a)}$

Multiply $5$ by $2$ .

$15 a^{4} \sqrt{10 a \cdot a}$

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

$15 a^{4} \sqrt{10 (a^{1} a)}$

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

$15 a^{4} \sqrt{10 (a^{1} a^{1})}$

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

$15 a^{4} \sqrt{10 a^{1 + 1}}$

Add $1$ and $1$ .

$15 a^{4} \sqrt{10 a^{2}}$

Step 7

Reorder $10$ and $a^{2}$ .

$15 a^{4} \sqrt{a^{2} \cdot 10}$

Step 8

Pull terms out from under the radical.

$15 a^{4} (a \sqrt{10})$

Step 9

Multiply $a^{4}$ by $a$ by adding the exponents.

Move $a$ .

$15 (a \cdot a^{4}) \sqrt{10}$

Multiply $a$ by $a^{4}$ .

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

$15 (a^{1} a^{4}) \sqrt{10}$

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

$15 a^{1 + 4} \sqrt{10}$

Add $1$ and $4$ .

$15 a^{5} \sqrt{10}$

Simplify 3 square root of 5a^3* square root of 50a^7

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