Simplify square root of 7x^4* square root of 8x^7

$\sqrt{7 x^{4}} \cdot \sqrt{8 x^{7}}$

Step 1

Rewrite $7 x^{4}$ as ${(x^{2})}^{2} \cdot 7$ .

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

$\sqrt{7 {(x^{2})}^{2}} \cdot \sqrt{8 x^{7}}$

Reorder $7$ and ${(x^{2})}^{2}$ .

$\sqrt{{(x^{2})}^{2} \cdot 7} \cdot \sqrt{8 x^{7}}$

Step 2

Pull terms out from under the radical.

$x^{2} \sqrt{7} \cdot \sqrt{8 x^{7}}$

Step 3

Rewrite $8 x^{7}$ as ${(2 x^{3})}^{2} \cdot (2 x)$ .

Factor $4$ out of $8$ .

$x^{2} \sqrt{7} \cdot \sqrt{4 (2) x^{7}}$

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

$x^{2} \sqrt{7} \cdot \sqrt{2^{2} \cdot 2 x^{7}}$

Factor out $x^{6}$ .

$x^{2} \sqrt{7} \cdot \sqrt{2^{2} \cdot 2 (x^{6} x)}$

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

$x^{2} \sqrt{7} \cdot \sqrt{2^{2} \cdot 2 ({(x^{3})}^{2} x)}$

Move $2$ .

$x^{2} \sqrt{7} \cdot \sqrt{2^{2} {(x^{3})}^{2} \cdot 2 x}$

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

$x^{2} \sqrt{7} \cdot \sqrt{{(2 x^{3})}^{2} \cdot 2 x}$

Add parentheses.

$x^{2} \sqrt{7} \cdot \sqrt{{(2 x^{3})}^{2} \cdot (2 x)}$

Step 4

Pull terms out from under the radical.

$x^{2} \sqrt{7} \cdot (2 x^{3} \sqrt{2 x})$

Step 5

Multiply $x^{2}$ by $x^{3}$ by adding the exponents.

Move $x^{3}$ .

$x^{3} x^{2} \sqrt{7} \cdot (2 \sqrt{2 x})$

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

$x^{3 + 2} \sqrt{7} \cdot (2 \sqrt{2 x})$

Add $3$ and $2$ .

$x^{5} \sqrt{7} \cdot (2 \sqrt{2 x})$

Step 6

Multiply $x^{5} \sqrt{7} (2 \sqrt{2 x})$ .

Combine using the product rule for radicals.

$x^{5} (2 \sqrt{7 (2 x)})$

Multiply $7$ by $2$ .

$x^{5} (2 \sqrt{14 x})$

Step 7

Rewrite using the commutative property of multiplication.

$2 x^{5} \sqrt{14 x}$

Simplify square root of 7x^4* square root of 8x^7

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