Basic Math Examples

$\sqrt{3 a^{2} b^{2}} \cdot \sqrt{9 a^{4} b^{6}} \cdot \sqrt{12 a^{2}}$

Step 1

Rewrite $3 a^{2} b^{2}$ as ${(a b)}^{2} \cdot 3$ .

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Step 1.1

Move $3$ .

$\sqrt{a^{2} b^{2} \cdot 3} \cdot \sqrt{9 a^{4} b^{6}} \cdot \sqrt{12 a^{2}}$

Step 1.2

Rewrite $a^{2} b^{2}$ as ${(a b)}^{2}$ .

$\sqrt{{(a b)}^{2} \cdot 3} \cdot \sqrt{9 a^{4} b^{6}} \cdot \sqrt{12 a^{2}}$

Step 2

Pull terms out from under the radical.

$a b \sqrt{3} \cdot \sqrt{9 a^{4} b^{6}} \cdot \sqrt{12 a^{2}}$

Step 3

Rewrite $9 a^{4} b^{6}$ as ${(3 a^{2} b^{3})}^{2}$ .

$a b \sqrt{3} \cdot \sqrt{{(3 a^{2} b^{3})}^{2}} \cdot \sqrt{12 a^{2}}$

Step 4

Pull terms out from under the radical, assuming positive real numbers.

$a b \sqrt{3} \cdot (3 a^{2} b^{3}) \cdot \sqrt{12 a^{2}}$

Step 5

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

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Step 5.1

Move $a^{2}$ .

$a^{2} a b \sqrt{3} \cdot (3 b^{3}) \cdot \sqrt{12 a^{2}}$

Step 5.2

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

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Step 5.2.1

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

$a^{2} a^{1} b \sqrt{3} \cdot (3 b^{3}) \cdot \sqrt{12 a^{2}}$

Step 5.2.2

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

$a^{2 + 1} b \sqrt{3} \cdot (3 b^{3}) \cdot \sqrt{12 a^{2}}$

Step 5.3

Add $2$ and $1$ .

$a^{3} b \sqrt{3} \cdot (3 b^{3}) \cdot \sqrt{12 a^{2}}$

Step 6

Multiply $b$ by $b^{3}$ by adding the exponents.

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Step 6.1

Move $b^{3}$ .

$a^{3} (b^{3} b) \sqrt{3} \cdot 3 \cdot \sqrt{12 a^{2}}$

Step 6.2

Multiply $b^{3}$ by $b$ .

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Step 6.2.1

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

$a^{3} (b^{3} b^{1}) \sqrt{3} \cdot 3 \cdot \sqrt{12 a^{2}}$

Step 6.2.2

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

$a^{3} b^{3 + 1} \sqrt{3} \cdot 3 \cdot \sqrt{12 a^{2}}$

Step 6.3

Add $3$ and $1$ .

$a^{3} b^{4} \sqrt{3} \cdot 3 \cdot \sqrt{12 a^{2}}$

Step 7

Move $3$ to the left of $a^{3} b^{4} \sqrt{3}$ .

$3 \cdot (a^{3} b^{4} \sqrt{3}) \cdot \sqrt{12 a^{2}}$

Step 8

Rewrite $12 a^{2}$ as ${(2 a)}^{2} \cdot 3$ .

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Step 8.1

Factor $4$ out of $12$ .

$3 a^{3} b^{4} \sqrt{3} \cdot \sqrt{4 (3) a^{2}}$

Step 8.2

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

$3 a^{3} b^{4} \sqrt{3} \cdot \sqrt{2^{2} \cdot 3 a^{2}}$

Step 8.3

Move $3$ .

$3 a^{3} b^{4} \sqrt{3} \cdot \sqrt{2^{2} a^{2} \cdot 3}$

Step 8.4

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

$3 a^{3} b^{4} \sqrt{3} \cdot \sqrt{{(2 a)}^{2} \cdot 3}$

Step 9

Pull terms out from under the radical.

$3 a^{3} b^{4} \sqrt{3} \cdot (2 a \sqrt{3})$

Step 10

Multiply $a^{3}$ by $a$ by adding the exponents.

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Step 10.1

Move $a$ .

$3 (a \cdot a^{3}) b^{4} \sqrt{3} \cdot (2 \sqrt{3})$

Step 10.2

Multiply $a$ by $a^{3}$ .

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Step 10.2.1

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

$3 (a^{1} a^{3}) b^{4} \sqrt{3} \cdot (2 \sqrt{3})$

Step 10.2.2

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

$3 a^{1 + 3} b^{4} \sqrt{3} \cdot (2 \sqrt{3})$

Step 10.3

Add $1$ and $3$ .

$3 a^{4} b^{4} \sqrt{3} \cdot (2 \sqrt{3})$

Step 11

Multiply $3 a^{4} b^{4} \sqrt{3} (2 \sqrt{3})$ .

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Step 11.1

Multiply $2$ by $3$ .

$6 a^{4} b^{4} \sqrt{3} \sqrt{3}$

Step 11.2

Raise $\sqrt{3}$ to the power of $1$ .

$6 a^{4} b^{4} ({\sqrt{3}}^{1} \sqrt{3})$

Step 11.3

Raise $\sqrt{3}$ to the power of $1$ .

$6 a^{4} b^{4} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 11.4

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

$6 a^{4} b^{4} {\sqrt{3}}^{1 + 1}$

Step 11.5

Add $1$ and $1$ .

$6 a^{4} b^{4} {\sqrt{3}}^{2}$

Step 12

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 12.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$6 a^{4} b^{4} {(3^{\frac{1}{2}})}^{2}$

Step 12.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$6 a^{4} b^{4} \cdot 3^{\frac{1}{2} \cdot 2}$

Step 12.3

Combine $\frac{1}{2}$ and $2$ .

$6 a^{4} b^{4} \cdot 3^{\frac{2}{2}}$

Step 12.4

Cancel the common factor of $2$ .

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Step 12.4.1

Cancel the common factor.

$6 a^{4} b^{4} \cdot 3^{\frac{2}{2}}$

Step 12.4.2

Rewrite the expression.

$6 a^{4} b^{4} \cdot 3^{1}$

Step 12.5

Evaluate the exponent.

$6 a^{4} b^{4} \cdot 3$

Step 13

Multiply $3$ by $6$ .

$18 a^{4} b^{4}$