Calculus Examples

$\sqrt{20 w^{5} x^{3}} \sqrt{5 w^{7} x^{2}}$

Step 1

Rewrite $20 w^{5} x^{3}$ as ${(2 w^{2} x)}^{2} \cdot (5 w x)$ .

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Step 1.1

Factor $4$ out of $20$ .

$\sqrt{4 (5) w^{5} x^{3}} \sqrt{5 w^{7} x^{2}}$

Step 1.2

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

$\sqrt{2^{2} \cdot 5 w^{5} x^{3}} \sqrt{5 w^{7} x^{2}}$

Step 1.3

Factor out $w^{4}$ .

$\sqrt{2^{2} \cdot 5 (w^{4} w) x^{3}} \sqrt{5 w^{7} x^{2}}$

Step 1.4

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

$\sqrt{2^{2} \cdot 5 ({(w^{2})}^{2} w) x^{3}} \sqrt{5 w^{7} x^{2}}$

Step 1.5

Factor out $x^{2}$ .

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

Step 1.6

Move $w$ .

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

Step 1.7

Move $5$ .

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

Step 1.8

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

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

Step 1.9

Add parentheses.

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

Step 1.10

Add parentheses.

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

Step 2

Pull terms out from under the radical.

$2 w^{2} x \sqrt{5 w x} \sqrt{5 w^{7} x^{2}}$

Step 3

Rewrite $5 w^{7} x^{2}$ as ${(w^{3} x)}^{2} \cdot (5 w)$ .

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Step 3.1

Factor out $w^{6}$ .

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

Step 3.2

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

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

Step 3.3

Move $w$ .

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

Step 3.4

Move $5$ .

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

Step 3.5

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

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

Step 3.6

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $w^{3}$ .

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

Step 5.2

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

$2 w^{3 + 2} x \sqrt{5 w x} (x \sqrt{5 w})$

Step 5.3

Add $3$ and $2$ .

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

Step 6

Multiply $x$ by $x$ by adding the exponents.

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Step 6.1

Move $x$ .

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

Step 6.2

Multiply $x$ by $x$ .

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

Step 7

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

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $5$ by $5$ .

$2 w^{5} x^{2} \sqrt{25 w (w x)}$

Step 7.3

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

$2 w^{5} x^{2} \sqrt{25 (w^{1} w) x}$

Step 7.4

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

$2 w^{5} x^{2} \sqrt{25 (w^{1} w^{1}) x}$

Step 7.5

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

$2 w^{5} x^{2} \sqrt{25 w^{1 + 1} x}$

Step 7.6

Add $1$ and $1$ .

$2 w^{5} x^{2} \sqrt{25 w^{2} x}$

Step 8

Rewrite $25 w^{2} x$ as ${(5 w)}^{2} x$ .

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Step 8.1

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

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

Step 8.2

Rewrite $5^{2} w^{2}$ as ${(5 w)}^{2}$ .

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

Step 9

Pull terms out from under the radical.

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

Step 10

Multiply $w^{5}$ by $w$ by adding the exponents.

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Step 10.1

Move $w$ .

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

Step 10.2

Multiply $w$ by $w^{5}$ .

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Step 10.2.1

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

$2 (w^{1} w^{5}) x^{2} (5 \sqrt{x})$

Step 10.2.2

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

$2 w^{1 + 5} x^{2} (5 \sqrt{x})$

Step 10.3

Add $1$ and $5$ .

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

Step 11

Multiply $5$ by $2$ .

$10 w^{6} x^{2} \sqrt{x}$