Simplify (3 square root of 2p^3q)(5p square root of 8pq)

$(3 \sqrt{2 p^{3} q}) (5 p \sqrt{8 p q})$

Step 1

Rewrite $2 p^{3} q$ as $p^{2} \cdot (2 p q)$ .

Factor out $p^{2}$ .

$3 \sqrt{2 (p^{2} p) q} (5 p \sqrt{8 p q})$

Reorder $2$ and $p^{2}$ .

$3 \sqrt{p^{2} \cdot 2 p q} (5 p \sqrt{8 p q})$

Add parentheses.

$3 \sqrt{p^{2} \cdot 2 (p q)} (5 p \sqrt{8 p q})$

Add parentheses.

$3 \sqrt{p^{2} \cdot (2 p q)} (5 p \sqrt{8 p q})$

Step 2

Pull terms out from under the radical.

$3 (p \sqrt{2 p q}) (5 p \sqrt{8 p q})$

Step 3

Multiply $p$ by $p$ by adding the exponents.

Move $p$ .

$3 (p \cdot p \sqrt{2 p q}) (5 \sqrt{8 p q})$

Multiply $p$ by $p$ .

$3 (p^{2} \sqrt{2 p q}) (5 \sqrt{8 p q})$

Step 4

Rewrite $8 p q$ as $2^{2} \cdot (2 p q)$ .

Factor $4$ out of $8$ .

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

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

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

Add parentheses.

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

Add parentheses.

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

Step 5

Pull terms out from under the radical.

$3 p^{2} \sqrt{2 p q} (5 (2 \sqrt{2 p q}))$

Step 6

Multiply $2$ by $5$ .

$3 p^{2} \sqrt{2 p q} (10 \sqrt{2 p q})$

Step 7

Multiply $3 p^{2} \sqrt{2 p q} (10 \sqrt{2 p q})$ .

Multiply $10$ by $3$ .

$30 p^{2} \sqrt{2 p q} \sqrt{2 p q}$

Raise $\sqrt{2 p q}$ to the power of $1$ .

$30 p^{2} ({\sqrt{2 p q}}^{1} \sqrt{2 p q})$

Raise $\sqrt{2 p q}$ to the power of $1$ .

$30 p^{2} ({\sqrt{2 p q}}^{1} {\sqrt{2 p q}}^{1})$

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

$30 p^{2} {\sqrt{2 p q}}^{1 + 1}$

Add $1$ and $1$ .

$30 p^{2} {\sqrt{2 p q}}^{2}$

Step 8

Rewrite ${\sqrt{2 p q}}^{2}$ as $2 p q$ .

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

$30 p^{2} {({(2 p q)}^{\frac{1}{2}})}^{2}$

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

$30 p^{2} {(2 p q)}^{\frac{1}{2} \cdot 2}$

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

$30 p^{2} {(2 p q)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$30 p^{2} {(2 p q)}^{\frac{2}{2}}$

Rewrite the expression.

$30 p^{2} {(2 p q)}^{1}$

Simplify.

$30 p^{2} (2 p q)$

Step 9

Multiply $p^{2}$ by $p$ by adding the exponents.

Move $p$ .

$30 (p \cdot p^{2}) (2 q)$

Multiply $p$ by $p^{2}$ .

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

$30 (p^{1} p^{2}) (2 q)$

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

$30 p^{1 + 2} (2 q)$

Add $1$ and $2$ .

$30 p^{3} (2 q)$

Step 10

Rewrite using the commutative property of multiplication.

$30 \cdot 2 p^{3} q$

Step 11

Multiply $30$ by $2$ .

$60 p^{3} q$

Simplify (3 square root of 2p^3q)(5p square root of 8pq)

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