Basic Math Examples

${(2 p^{- 2} q)}^{3} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(2 \frac{1}{p^{2}} q)}^{3} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 2

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

${(\frac{2}{p^{2}} q)}^{3} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 3

Combine $\frac{2}{p^{2}}$ and $q$ .

${(\frac{2 q}{p^{2}})}^{3} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $\frac{2 q}{p^{2}}$ .

$\frac{{(2 q)}^{3}}{{(p^{2})}^{3}} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 4.2

Apply the product rule to $2 q$ .

$\frac{2^{3} q^{3}}{{(p^{2})}^{3}} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 5

Simplify terms.

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Step 5.1

Raise $2$ to the power of $3$ .

$\frac{8 q^{3}}{{(p^{2})}^{3}} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 5.2

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

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Step 5.2.1

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

$\frac{8 q^{3}}{p^{2 \cdot 3}} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 5.2.2

Multiply $2$ by $3$ .

$\frac{8 q^{3}}{p^{6}} (\frac{1}{2} \cdot (p^{2} q) - 1)$

Step 5.3

Multiply $\frac{1}{2} (p^{2} q)$ .

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Step 5.3.1

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

$\frac{8 q^{3}}{p^{6}} (\frac{p^{2}}{2} q - 1)$

Step 5.3.2

Combine $\frac{p^{2}}{2}$ and $q$ .

$\frac{8 q^{3}}{p^{6}} (\frac{p^{2} q}{2} - 1)$

Step 5.4

Multiply $\frac{8 q^{3}}{p^{6}}$ by $\frac{p^{2} q}{2} - 1$ .

$\frac{8 q^{3} (\frac{p^{2} q}{2} - 1)}{p^{6}}$

Step 6

Simplify the numerator.

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Step 6.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{8 q^{3} (\frac{p^{2} q}{2} - 1 \cdot \frac{2}{2})}{p^{6}}$

Step 6.2

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

$\frac{8 q^{3} (\frac{p^{2} q}{2} + \frac{- 1 \cdot 2}{2})}{p^{6}}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{8 q^{3} \frac{p^{2} q - 1 \cdot 2}{2}}{p^{6}}$

Step 6.4

Multiply $- 1$ by $2$ .

$\frac{8 q^{3} \frac{p^{2} q - 2}{2}}{p^{6}}$

Step 6.5

Combine exponents.

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Step 6.5.1

Combine $8$ and $\frac{p^{2} q - 2}{2}$ .

$\frac{q^{3} \frac{8 (p^{2} q - 2)}{2}}{p^{6}}$

Step 6.5.2

Combine $q^{3}$ and $\frac{8 (p^{2} q - 2)}{2}$ .

$\frac{\frac{q^{3} (8 (p^{2} q - 2))}{2}}{p^{6}}$

Step 6.6

Remove unnecessary parentheses.

$\frac{\frac{q^{3} \cdot 8 (p^{2} q - 2)}{2}}{p^{6}}$

Step 6.7

Reduce the expression by cancelling the common factors.

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Step 6.7.1

Reduce the expression $\frac{q^{3} \cdot 8 (p^{2} q - 2)}{2}$ by cancelling the common factors.

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Step 6.7.1.1

Factor $2$ out of $q^{3} \cdot 8 (p^{2} q - 2)$ .

$\frac{\frac{2 (q^{3} \cdot 4 (p^{2} q - 2))}{2}}{p^{6}}$

Step 6.7.1.2

Factor $2$ out of $2$ .

$\frac{\frac{2 (q^{3} \cdot 4 (p^{2} q - 2))}{2 (1)}}{p^{6}}$

Step 6.7.1.3

Cancel the common factor.

$\frac{\frac{2 (q^{3} \cdot 4 (p^{2} q - 2))}{2 \cdot 1}}{p^{6}}$

Step 6.7.1.4

Rewrite the expression.

$\frac{\frac{q^{3} \cdot 4 (p^{2} q - 2)}{1}}{p^{6}}$

Step 6.7.2

Divide $q^{3} \cdot 4 (p^{2} q - 2)$ by $1$ .

$\frac{q^{3} \cdot 4 (p^{2} q - 2)}{p^{6}}$

Step 7

Move $4$ to the left of $q^{3}$ .

$\frac{4 q^{3} (p^{2} q - 2)}{p^{6}}$