Calculus Examples

$\sqrt{5 x^{9} u^{2}} \sqrt{20 x u^{3}}$

Step 1

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

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Step 1.1

Factor out $x^{8}$ .

$\sqrt{5 (x^{8} x) u^{2}} \sqrt{20 x u^{3}}$

Step 1.2

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

$\sqrt{5 ({(x^{4})}^{2} x) u^{2}} \sqrt{20 x u^{3}}$

Step 1.3

Move $x$ .

$\sqrt{5 {(x^{4})}^{2} u^{2} x} \sqrt{20 x u^{3}}$

Step 1.4

Move $5$ .

$\sqrt{{(x^{4})}^{2} u^{2} \cdot 5 x} \sqrt{20 x u^{3}}$

Step 1.5

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

$\sqrt{{(x^{4} u)}^{2} \cdot 5 x} \sqrt{20 x u^{3}}$

Step 1.6

Add parentheses.

$\sqrt{{(x^{4} u)}^{2} \cdot (5 x)} \sqrt{20 x u^{3}}$

Step 2

Pull terms out from under the radical.

$x^{4} u \sqrt{5 x} \sqrt{20 x u^{3}}$

Step 3

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

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Step 3.1

Factor $4$ out of $20$ .

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

Step 3.2

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

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

Step 3.3

Factor out $u^{2}$ .

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

Step 3.4

Move $x$ .

$x^{4} u \sqrt{5 x} \sqrt{2^{2} \cdot 5 u^{2} x u}$

Step 3.5

Move $5$ .

$x^{4} u \sqrt{5 x} \sqrt{2^{2} u^{2} \cdot 5 x u}$

Step 3.6

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

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

Step 3.7

Add parentheses.

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

Step 3.8

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

Rewrite using the commutative property of multiplication.

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

Step 6

Multiply $u$ by $u$ by adding the exponents.

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Step 6.1

Move $u$ .

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

Step 6.2

Multiply $u$ by $u$ .

$2 x^{4} u^{2} \sqrt{5 x} \sqrt{5 x u}$

Step 7

Multiply $2 x^{4} u^{2} \sqrt{5 x} \sqrt{5 x u}$ .

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Step 7.1

Combine using the product rule for radicals.

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

Step 7.2

Multiply $5$ by $5$ .

$2 x^{4} u^{2} \sqrt{25 x u x}$

Step 7.3

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

$2 x^{4} u^{2} \sqrt{25 u (x^{1} x)}$

Step 7.4

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

$2 x^{4} u^{2} \sqrt{25 u (x^{1} x^{1})}$

Step 7.5

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

$2 x^{4} u^{2} \sqrt{25 u x^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$2 x^{4} u^{2} \sqrt{25 u x^{2}}$

Step 8

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

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Step 8.1

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

$2 x^{4} u^{2} \sqrt{5^{2} u x^{2}}$

Step 8.2

Move $u$ .

$2 x^{4} u^{2} \sqrt{5^{2} x^{2} u}$

Step 8.3

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

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

Step 9

Pull terms out from under the radical.

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

Step 10

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

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Step 10.1

Move $x$ .

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

Step 10.2

Multiply $x$ by $x^{4}$ .

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Step 10.2.1

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

$2 (x^{1} x^{4}) u^{2} (5 \sqrt{u})$

Step 10.2.2

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

$2 x^{1 + 4} u^{2} (5 \sqrt{u})$

Step 10.3

Add $1$ and $4$ .

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

Step 11

Multiply $5$ by $2$ .

$10 x^{5} u^{2} \sqrt{u}$