Basic Math Examples

$\sqrt{20 x^{4} u^{5}} \sqrt{5 x^{3} u^{9}}$

Step 1

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

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Step 1.1

Factor $4$ out of $20$ .

$\sqrt{4 (5) x^{4} u^{5}} \sqrt{5 x^{3} u^{9}}$

Step 1.2

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

$\sqrt{2^{2} \cdot 5 x^{4} u^{5}} \sqrt{5 x^{3} u^{9}}$

Step 1.3

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

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

Step 1.4

Factor out $u^{4}$ .

$\sqrt{2^{2} \cdot 5 {(x^{2})}^{2} (u^{4} u)} \sqrt{5 x^{3} u^{9}}$

Step 1.5

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

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

Step 1.6

Move $5$ .

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

Step 1.7

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

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

Step 1.8

Add parentheses.

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

Step 2

Pull terms out from under the radical.

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

Step 3

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

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Step 3.1

Factor out $x^{2}$ .

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

Step 3.2

Factor out $u^{8}$ .

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

Step 3.3

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

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

Step 3.4

Move $x$ .

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

Step 3.5

Move $5$ .

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

Step 3.6

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

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

Step 3.7

Add parentheses.

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

Step 3.8

Add parentheses.

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

Step 4

Pull terms out from under the radical.

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

Step 5

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

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Step 5.1

Move $x$ .

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

Step 5.2

Multiply $x$ by $x^{2}$ .

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 5.3

Add $1$ and $2$ .

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

Step 6

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

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Step 6.1

Move $u^{4}$ .

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

Step 6.2

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

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

Step 6.3

Add $4$ and $2$ .

$2 x^{3} u^{6} \sqrt{5 u} \sqrt{5 x u}$

Step 7

Multiply $2 x^{3} u^{6} \sqrt{5 u} \sqrt{5 x u}$ .

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Step 7.1

Combine using the product rule for radicals.

$2 x^{3} u^{6} \sqrt{5 x u (5 u)}$

Step 7.2

Multiply $5$ by $5$ .

$2 x^{3} u^{6} \sqrt{25 x u \cdot u}$

Step 7.3

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

$2 x^{3} u^{6} \sqrt{25 x (u^{1} u)}$

Step 7.4

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

$2 x^{3} u^{6} \sqrt{25 x (u^{1} u^{1})}$

Step 7.5

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

$2 x^{3} u^{6} \sqrt{25 x u^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$2 x^{3} u^{6} \sqrt{25 x u^{2}}$

Step 8

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

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Step 8.1

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

$2 x^{3} u^{6} \sqrt{5^{2} x u^{2}}$

Step 8.2

Move $x$ .

$2 x^{3} u^{6} \sqrt{5^{2} u^{2} x}$

Step 8.3

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

$2 x^{3} u^{6} \sqrt{{(5 u)}^{2} x}$

Step 9

Pull terms out from under the radical.

$2 x^{3} u^{6} (5 u \sqrt{x})$

Step 10

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

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Step 10.1

Move $u$ .

$2 x^{3} (u \cdot u^{6}) (5 \sqrt{x})$

Step 10.2

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

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Step 10.2.1

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

$2 x^{3} (u^{1} u^{6}) (5 \sqrt{x})$

Step 10.2.2

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

$2 x^{3} u^{1 + 6} (5 \sqrt{x})$

Step 10.3

Add $1$ and $6$ .

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

Step 11

Multiply $5$ by $2$ .

$10 x^{3} u^{7} \sqrt{x}$