Basic Math Examples

$\sqrt{12 m^{5}} \cdot \sqrt{12 m^{5}}$

Step 1

Rewrite $12 m^{5}$ as ${(2 m^{2})}^{2} \cdot (3 m)$ .

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Step 1.1

Factor $4$ out of $12$ .

$\sqrt{4 (3) m^{5}} \cdot \sqrt{12 m^{5}}$

Step 1.2

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

$\sqrt{2^{2} \cdot 3 m^{5}} \cdot \sqrt{12 m^{5}}$

Step 1.3

Factor out $m^{4}$ .

$\sqrt{2^{2} \cdot 3 (m^{4} m)} \cdot \sqrt{12 m^{5}}$

Step 1.4

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

$\sqrt{2^{2} \cdot 3 ({(m^{2})}^{2} m)} \cdot \sqrt{12 m^{5}}$

Step 1.5

Move $3$ .

$\sqrt{2^{2} {(m^{2})}^{2} \cdot 3 m} \cdot \sqrt{12 m^{5}}$

Step 1.6

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

$\sqrt{{(2 m^{2})}^{2} \cdot 3 m} \cdot \sqrt{12 m^{5}}$

Step 1.7

Add parentheses.

$\sqrt{{(2 m^{2})}^{2} \cdot (3 m)} \cdot \sqrt{12 m^{5}}$

Step 2

Pull terms out from under the radical.

$2 m^{2} \sqrt{3 m} \cdot \sqrt{12 m^{5}}$

Step 3

Rewrite $12 m^{5}$ as ${(2 m^{2})}^{2} \cdot (3 m)$ .

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Step 3.1

Factor $4$ out of $12$ .

$2 m^{2} \sqrt{3 m} \cdot \sqrt{4 (3) m^{5}}$

Step 3.2

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

$2 m^{2} \sqrt{3 m} \cdot \sqrt{2^{2} \cdot 3 m^{5}}$

Step 3.3

Factor out $m^{4}$ .

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

Step 3.4

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

$2 m^{2} \sqrt{3 m} \cdot \sqrt{2^{2} \cdot 3 ({(m^{2})}^{2} m)}$

Step 3.5

Move $3$ .

$2 m^{2} \sqrt{3 m} \cdot \sqrt{2^{2} {(m^{2})}^{2} \cdot 3 m}$

Step 3.6

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

$2 m^{2} \sqrt{3 m} \cdot \sqrt{{(2 m^{2})}^{2} \cdot 3 m}$

Step 3.7

Add parentheses.

$2 m^{2} \sqrt{3 m} \cdot \sqrt{{(2 m^{2})}^{2} \cdot (3 m)}$

Step 4

Pull terms out from under the radical.

$2 m^{2} \sqrt{3 m} \cdot (2 m^{2} \sqrt{3 m})$

Step 5

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

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Step 5.1

Move $m^{2}$ .

$2 (m^{2} m^{2}) \sqrt{3 m} \cdot (2 \sqrt{3 m})$

Step 5.2

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

$2 m^{2 + 2} \sqrt{3 m} \cdot (2 \sqrt{3 m})$

Step 5.3

Add $2$ and $2$ .

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

Step 6

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

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Step 6.1

Multiply $2$ by $2$ .

$4 m^{4} \sqrt{3 m} \sqrt{3 m}$

Step 6.2

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

$4 m^{4} ({\sqrt{3 m}}^{1} \sqrt{3 m})$

Step 6.3

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

$4 m^{4} ({\sqrt{3 m}}^{1} {\sqrt{3 m}}^{1})$

Step 6.4

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

$4 m^{4} {\sqrt{3 m}}^{1 + 1}$

Step 6.5

Add $1$ and $1$ .

$4 m^{4} {\sqrt{3 m}}^{2}$

Step 7

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

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Step 7.1

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

$4 m^{4} {({(3 m)}^{\frac{1}{2}})}^{2}$

Step 7.2

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

$4 m^{4} {(3 m)}^{\frac{1}{2} \cdot 2}$

Step 7.3

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

$4 m^{4} {(3 m)}^{\frac{2}{2}}$

Step 7.4

Cancel the common factor of $2$ .

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Step 7.4.1

Cancel the common factor.

$4 m^{4} {(3 m)}^{\frac{2}{2}}$

Step 7.4.2

Rewrite the expression.

$4 m^{4} {(3 m)}^{1}$

Step 7.5

Simplify.

$4 m^{4} (3 m)$

Step 8

Rewrite using the commutative property of multiplication.

$4 \cdot 3 m^{4} m$

Step 9

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

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Step 9.1

Move $m$ .

$4 \cdot 3 (m \cdot m^{4})$

Step 9.2

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

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Step 9.2.1

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

$4 \cdot 3 (m^{1} m^{4})$

Step 9.2.2

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

$4 \cdot 3 m^{1 + 4}$

Step 9.3

Add $1$ and $4$ .

$4 \cdot 3 m^{5}$

Step 10

Multiply $4$ by $3$ .

$12 m^{5}$