Basic Math Examples

$2 {(m^{2} n \div m)}^{6} \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 1

Rewrite the division as a fraction.

$2 {(\frac{m^{2} n}{m})}^{6} \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 2

Cancel the common factor of $m^{2}$ and $m$ .

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Step 2.1

Factor $m$ out of $m^{2} n$ .

$2 {(\frac{m (m n)}{m})}^{6} \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

$2 {(\frac{m (m n)}{m^{1}})}^{6} \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 2.2.2

Factor $m$ out of $m^{1}$ .

$2 {(\frac{m (m n)}{m \cdot 1})}^{6} \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 2.2.3

Cancel the common factor.

$2 {(\frac{m (m n)}{m \cdot 1})}^{6} \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 2.2.4

Rewrite the expression.

$2 {(\frac{m n}{1})}^{6} \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 2.2.5

Divide $m n$ by $1$ .

$2 {(m n)}^{6} \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 3

Apply basic rules of exponents.

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Step 3.1

Apply the product rule to $m n$ .

$2 (m^{6} n^{6}) \cdot {(3^{\frac{1}{2}} n^{3})}^{2}$

Step 3.2

Apply the product rule to $3^{\frac{1}{2}} n^{3}$ .

$2 m^{6} n^{6} \cdot ({(3^{\frac{1}{2}})}^{2} {(n^{3})}^{2})$

Step 3.3

Multiply the exponents in ${(3^{\frac{1}{2}})}^{2}$ .

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Step 3.3.1

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

$2 m^{6} n^{6} \cdot (3^{\frac{1}{2} \cdot 2} {(n^{3})}^{2})$

Step 3.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1

Cancel the common factor.

$2 m^{6} n^{6} \cdot (3^{\frac{1}{2} \cdot 2} {(n^{3})}^{2})$

Step 3.3.2.2

Rewrite the expression.

$2 m^{6} n^{6} \cdot (3^{1} {(n^{3})}^{2})$

Step 4

Evaluate the exponent.

$2 m^{6} n^{6} \cdot (3 {(n^{3})}^{2})$

Step 5

Multiply the exponents in ${(n^{3})}^{2}$ .

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Step 5.1

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

$2 m^{6} n^{6} \cdot (3 n^{3 \cdot 2})$

Step 5.2

Multiply $3$ by $2$ .

$2 m^{6} n^{6} \cdot (3 n^{6})$

Step 6

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

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Step 6.1

Move $n^{6}$ .

$2 m^{6} (n^{6} n^{6}) \cdot 3$

Step 6.2

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

$2 m^{6} n^{6 + 6} \cdot 3$

Step 6.3

Add $6$ and $6$ .

$2 m^{6} n^{12} \cdot 3$

Step 7

Multiply $3$ by $2$ .

$6 m^{6} n^{12}$