Calculus Examples

$\sqrt{10 x^{2} w^{5}} \sqrt{5 x^{3} w^{9}}$

Step 1

Rewrite $10 x^{2} w^{5}$ as ${(x w^{2})}^{2} \cdot (10 w)$ .

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Step 1.1

Factor out $w^{4}$ .

$\sqrt{10 x^{2} (w^{4} w)} \sqrt{5 x^{3} w^{9}}$

Step 1.2

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

$\sqrt{10 x^{2} ({(w^{2})}^{2} w)} \sqrt{5 x^{3} w^{9}}$

Step 1.3

Move $10$ .

$\sqrt{x^{2} {(w^{2})}^{2} \cdot 10 w} \sqrt{5 x^{3} w^{9}}$

Step 1.4

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

$\sqrt{{(x w^{2})}^{2} \cdot 10 w} \sqrt{5 x^{3} w^{9}}$

Step 1.5

Add parentheses.

$\sqrt{{(x w^{2})}^{2} \cdot (10 w)} \sqrt{5 x^{3} w^{9}}$

Step 2

Pull terms out from under the radical.

$x w^{2} \sqrt{10 w} \sqrt{5 x^{3} w^{9}}$

Step 3

Rewrite $5 x^{3} w^{9}$ as ${(x w^{4})}^{2} \cdot (5 x w)$ .

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Step 3.1

Factor out $x^{2}$ .

$x w^{2} \sqrt{10 w} \sqrt{5 (x^{2} x) w^{9}}$

Step 3.2

Factor out $w^{8}$ .

$x w^{2} \sqrt{10 w} \sqrt{5 (x^{2} x) (w^{8} w)}$

Step 3.3

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

$x w^{2} \sqrt{10 w} \sqrt{5 (x^{2} x) ({(w^{4})}^{2} w)}$

Step 3.4

Move $x$ .

$x w^{2} \sqrt{10 w} \sqrt{5 (x^{2}) {(w^{4})}^{2} x w}$

Step 3.5

Move $5$ .

$x w^{2} \sqrt{10 w} \sqrt{x^{2} {(w^{4})}^{2} \cdot 5 x w}$

Step 3.6

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

$x w^{2} \sqrt{10 w} \sqrt{{(x w^{4})}^{2} \cdot 5 x w}$

Step 3.7

Add parentheses.

$x w^{2} \sqrt{10 w} \sqrt{{(x w^{4})}^{2} \cdot 5 (x w)}$

Step 3.8

Add parentheses.

$x w^{2} \sqrt{10 w} \sqrt{{(x w^{4})}^{2} \cdot (5 x w)}$

Step 4

Pull terms out from under the radical.

$x w^{2} \sqrt{10 w} (x w^{4} \sqrt{5 x w})$

Step 5

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

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Step 5.1

Move $x$ .

$x \cdot x w^{2} \sqrt{10 w} (w^{4} \sqrt{5 x w})$

Step 5.2

Multiply $x$ by $x$ .

$x^{2} w^{2} \sqrt{10 w} (w^{4} \sqrt{5 x w})$

Step 6

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

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Step 6.1

Move $w^{4}$ .

$x^{2} (w^{4} w^{2}) \sqrt{10 w} \sqrt{5 x w}$

Step 6.2

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

$x^{2} w^{4 + 2} \sqrt{10 w} \sqrt{5 x w}$

Step 6.3

Add $4$ and $2$ .

$x^{2} w^{6} \sqrt{10 w} \sqrt{5 x w}$

Step 7

Multiply $x^{2} w^{6} \sqrt{10 w} \sqrt{5 x w}$ .

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Step 7.1

Combine using the product rule for radicals.

$x^{2} w^{6} \sqrt{5 x w (10 w)}$

Step 7.2

Multiply $10$ by $5$ .

$x^{2} w^{6} \sqrt{50 x w \cdot w}$

Step 7.3

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

$x^{2} w^{6} \sqrt{50 x (w^{1} w)}$

Step 7.4

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

$x^{2} w^{6} \sqrt{50 x (w^{1} w^{1})}$

Step 7.5

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

$x^{2} w^{6} \sqrt{50 x w^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$x^{2} w^{6} \sqrt{50 x w^{2}}$

Step 8

Rewrite $50 x w^{2}$ as ${(5 w)}^{2} \cdot (2 x)$ .

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Step 8.1

Factor $25$ out of $50$ .

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

Step 8.2

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

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

Step 8.3

Move $x$ .

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

Step 8.4

Move $2$ .

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

Step 8.5

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

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

Step 8.6

Add parentheses.

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

Step 9

Pull terms out from under the radical.

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

Step 10

Rewrite using the commutative property of multiplication.

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

Step 11

Multiply $w^{6}$ by $w$ by adding the exponents.

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Step 11.1

Move $w$ .

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

Step 11.2

Multiply $w$ by $w^{6}$ .

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Step 11.2.1

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

$5 x^{2} (w^{1} w^{6}) \sqrt{2 x}$

Step 11.2.2

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

$5 x^{2} w^{1 + 6} \sqrt{2 x}$

Step 11.3

Add $1$ and $6$ .

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