Simplify square root of 8x^3* square root of 2x^5

$\sqrt{8 x^{3}} \cdot \sqrt{2 x^{5}}$

Step 1

Rewrite $8 x^{3}$ as ${(2 x)}^{2} \cdot (2 x)$ .

Factor $4$ out of $8$ .

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

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

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

Factor out $x^{2}$ .

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

Move $2$ .

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

Rewrite $2^{2} x^{2}$ as ${(2 x)}^{2}$ .

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

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

Factor out $x^{4}$ .

$2 x \sqrt{2 x} \cdot \sqrt{2 (x^{4} x)}$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$2 x \sqrt{2 x} \cdot \sqrt{2 ({(x^{2})}^{2} x)}$

Reorder $2$ and ${(x^{2})}^{2}$ .

$2 x \sqrt{2 x} \cdot \sqrt{{(x^{2})}^{2} \cdot 2 x}$

Add parentheses.

$2 x \sqrt{2 x} \cdot \sqrt{{(x^{2})}^{2} \cdot (2 x)}$

Step 4

Pull terms out from under the radical.

$2 x \sqrt{2 x} \cdot (x^{2} \sqrt{2 x})$

Step 5

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

Move $x^{2}$ .

$2 (x^{2} x) \sqrt{2 x} \cdot \sqrt{2 x}$

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

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

$2 (x^{2} x^{1}) \sqrt{2 x} \cdot \sqrt{2 x}$

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

$2 x^{2 + 1} \sqrt{2 x} \cdot \sqrt{2 x}$

Add $2$ and $1$ .

$2 x^{3} \sqrt{2 x} \cdot \sqrt{2 x}$

Step 6

Multiply $2 x^{3} \sqrt{2 x} \sqrt{2 x}$ .

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

$2 x^{3} ({\sqrt{2 x}}^{1} \sqrt{2 x})$

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

$2 x^{3} ({\sqrt{2 x}}^{1} {\sqrt{2 x}}^{1})$

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

$2 x^{3} {\sqrt{2 x}}^{1 + 1}$

Add $1$ and $1$ .

$2 x^{3} {\sqrt{2 x}}^{2}$

Step 7

Rewrite ${\sqrt{2 x}}^{2}$ as $2 x$ .

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

$2 x^{3} {({(2 x)}^{\frac{1}{2}})}^{2}$

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

$2 x^{3} {(2 x)}^{\frac{1}{2} \cdot 2}$

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

$2 x^{3} {(2 x)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 x^{3} {(2 x)}^{\frac{2}{2}}$

Rewrite the expression.

$2 x^{3} {(2 x)}^{1}$

Simplify.

$2 x^{3} (2 x)$

Step 8

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x^{3} x$

Step 9

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

Move $x$ .

$2 \cdot 2 (x \cdot x^{3})$

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

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

$2 \cdot 2 (x^{1} x^{3})$

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

$2 \cdot 2 x^{1 + 3}$

Add $1$ and $3$ .

$2 \cdot 2 x^{4}$

Step 10

Multiply $2$ by $2$ .

$4 x^{4}$

Simplify square root of 8x^3* square root of 2x^5

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