Pre-Algebra Examples

$\sqrt{5 x^{8} w^{4}} \sqrt{20 x^{5} w^{8}}$

Step 1

Rewrite $5 x^{8} w^{4}$ as ${(x^{4} w^{2})}^{2} \cdot 5$ .

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Step 1.1

Rewrite $x^{8}$ as ${(x^{4})}^{2}$ .

$\sqrt{5 {(x^{4})}^{2} w^{4}} \sqrt{20 x^{5} w^{8}}$

Step 1.2

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

$\sqrt{5 {(x^{4})}^{2} {(w^{2})}^{2}} \sqrt{20 x^{5} w^{8}}$

Step 1.3

Move $5$ .

$\sqrt{{(x^{4})}^{2} {(w^{2})}^{2} \cdot 5} \sqrt{20 x^{5} w^{8}}$

Step 1.4

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

$\sqrt{{(x^{4} w^{2})}^{2} \cdot 5} \sqrt{20 x^{5} w^{8}}$

Step 2

Pull terms out from under the radical.

$x^{4} w^{2} \sqrt{5} \sqrt{20 x^{5} w^{8}}$

Step 3

Rewrite $20 x^{5} w^{8}$ as ${(2 x^{2} w^{4})}^{2} \cdot (5 x)$ .

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Step 3.1

Factor $4$ out of $20$ .

$x^{4} w^{2} \sqrt{5} \sqrt{4 (5) x^{5} w^{8}}$

Step 3.2

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

$x^{4} w^{2} \sqrt{5} \sqrt{2^{2} \cdot 5 x^{5} w^{8}}$

Step 3.3

Factor out $x^{4}$ .

$x^{4} w^{2} \sqrt{5} \sqrt{2^{2} \cdot 5 (x^{4} x) w^{8}}$

Step 3.4

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

$x^{4} w^{2} \sqrt{5} \sqrt{2^{2} \cdot 5 ({(x^{2})}^{2} x) w^{8}}$

Step 3.5

Rewrite $w^{8}$ as ${(w^{4})}^{2}$ .

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

Step 3.6

Move $x$ .

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

Step 3.7

Move $5$ .

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

Step 3.8

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

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

Step 3.9

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

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Step 5.1

Move $x^{2}$ .

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

Step 5.2

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

$x^{2 + 4} w^{2} \sqrt{5} (2 w^{4} \sqrt{5 x})$

Step 5.3

Add $2$ and $4$ .

$x^{6} w^{2} \sqrt{5} (2 w^{4} \sqrt{5 x})$

Step 6

Multiply $w^{2}$ by $w^{4}$ by adding the exponents.

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Step 6.1

Move $w^{4}$ .

$x^{6} (w^{4} w^{2}) \sqrt{5} (2 \sqrt{5 x})$

Step 6.2

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

$x^{6} w^{4 + 2} \sqrt{5} (2 \sqrt{5 x})$

Step 6.3

Add $4$ and $2$ .

$x^{6} w^{6} \sqrt{5} (2 \sqrt{5 x})$

Step 7

Multiply $x^{6} w^{6} \sqrt{5} (2 \sqrt{5 x})$ .

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Step 7.1

Combine using the product rule for radicals.

$x^{6} w^{6} (2 \sqrt{5 (5 x)})$

Step 7.2

Multiply $5$ by $5$ .

$x^{6} w^{6} (2 \sqrt{25 x})$

Step 8

Simplify the expression.

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Step 8.1

Rewrite using the commutative property of multiplication.

$2 x^{6} w^{6} \sqrt{25 x}$

Step 8.2

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

$2 x^{6} w^{6} \sqrt{5^{2} x}$

Step 9

Pull terms out from under the radical.

$2 x^{6} w^{6} (5 \sqrt{x})$

Step 10

Multiply $5$ by $2$ .

$10 x^{6} w^{6} \sqrt{x}$