Basic Math Examples

${(\frac{p^{- \frac{1}{2}} q^{- \frac{5}{2}}}{4^{- 1} p^{- 2} q^{- \frac{1}{5}}})}^{- 2}$

Step 1

Move $q^{- \frac{5}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{- \frac{1}{5}} q^{\frac{5}{2}}})}^{- 2}$

Step 2

Multiply $q^{- \frac{1}{5}}$ by $q^{\frac{5}{2}}$ by adding the exponents.

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Step 2.1

Move $q^{\frac{5}{2}}$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} (q^{\frac{5}{2}} q^{- \frac{1}{5}})})}^{- 2}$

Step 2.2

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

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{5}{2} - \frac{1}{5}}})}^{- 2}$

Step 2.3

To write $\frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{5}{2} \cdot \frac{5}{5} - \frac{1}{5}}})}^{- 2}$

Step 2.4

To write $- \frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{5}{2} \cdot \frac{5}{5} - \frac{1}{5} \cdot \frac{2}{2}}})}^{- 2}$

Step 2.5

Write each expression with a common denominator of $10$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.1

Multiply $\frac{5}{2}$ by $\frac{5}{5}$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{5 \cdot 5}{2 \cdot 5} - \frac{1}{5} \cdot \frac{2}{2}}})}^{- 2}$

Step 2.5.2

Multiply $2$ by $5$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{5 \cdot 5}{10} - \frac{1}{5} \cdot \frac{2}{2}}})}^{- 2}$

Step 2.5.3

Multiply $\frac{1}{5}$ by $\frac{2}{2}$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{5 \cdot 5}{10} - \frac{2}{5 \cdot 2}}})}^{- 2}$

Step 2.5.4

Multiply $5$ by $2$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{5 \cdot 5}{10} - \frac{2}{10}}})}^{- 2}$

Step 2.6

Combine the numerators over the common denominator.

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{5 \cdot 5 - 2}{10}}})}^{- 2}$

Step 2.7

Simplify the numerator.

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Step 2.7.1

Multiply $5$ by $5$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{25 - 2}{10}}})}^{- 2}$

Step 2.7.2

Subtract $2$ from $25$ .

${(\frac{p^{- \frac{1}{2}}}{4^{- 1} p^{- 2} q^{\frac{23}{10}}})}^{- 2}$

Step 3

Move $p^{- 2}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{p^{- \frac{1}{2}} p^{2}}{4^{- 1} q^{\frac{23}{10}}})}^{- 2}$

Step 4

Move $4^{- 1}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{p^{- \frac{1}{2}} p^{2} \cdot 4}{q^{\frac{23}{10}}})}^{- 2}$

Step 5

Multiply $p^{- \frac{1}{2}}$ by $p^{2}$ by adding the exponents.

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Step 5.1

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

${(\frac{p^{- \frac{1}{2} + 2} \cdot 4}{q^{\frac{23}{10}}})}^{- 2}$

Step 5.2

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

${(\frac{p^{- \frac{1}{2} + 2 \cdot \frac{2}{2}} \cdot 4}{q^{\frac{23}{10}}})}^{- 2}$

Step 5.3

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

${(\frac{p^{- \frac{1}{2} + \frac{2 \cdot 2}{2}} \cdot 4}{q^{\frac{23}{10}}})}^{- 2}$

Step 5.4

Combine the numerators over the common denominator.

${(\frac{p^{\frac{- 1 + 2 \cdot 2}{2}} \cdot 4}{q^{\frac{23}{10}}})}^{- 2}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $2$ by $2$ .

${(\frac{p^{\frac{- 1 + 4}{2}} \cdot 4}{q^{\frac{23}{10}}})}^{- 2}$

Step 5.5.2

Add $- 1$ and $4$ .

${(\frac{p^{\frac{3}{2}} \cdot 4}{q^{\frac{23}{10}}})}^{- 2}$

Step 6

Move $4$ to the left of $p^{\frac{3}{2}}$ .

${(\frac{4 \cdot p^{\frac{3}{2}}}{q^{\frac{23}{10}}})}^{- 2}$

Step 7

Change the sign of the exponent by rewriting the base as its reciprocal.

${(\frac{q^{\frac{23}{10}}}{4 p^{\frac{3}{2}}})}^{2}$

Step 8

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

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Step 8.1

Apply the product rule to $\frac{q^{\frac{23}{10}}}{4 p^{\frac{3}{2}}}$ .

$\frac{{(q^{\frac{23}{10}})}^{2}}{{(4 p^{\frac{3}{2}})}^{2}}$

Step 8.2

Apply the product rule to $4 p^{\frac{3}{2}}$ .

$\frac{{(q^{\frac{23}{10}})}^{2}}{4^{2} {(p^{\frac{3}{2}})}^{2}}$

Step 9

Multiply the exponents in ${(q^{\frac{23}{10}})}^{2}$ .

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Step 9.1

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

$\frac{q^{\frac{23}{10} \cdot 2}}{4^{2} {(p^{\frac{3}{2}})}^{2}}$

Step 9.2

Cancel the common factor of $2$ .

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Step 9.2.1

Factor $2$ out of $10$ .

$\frac{q^{\frac{23}{2 (5)} \cdot 2}}{4^{2} {(p^{\frac{3}{2}})}^{2}}$

Step 9.2.2

Cancel the common factor.

$\frac{q^{\frac{23}{2 \cdot 5} \cdot 2}}{4^{2} {(p^{\frac{3}{2}})}^{2}}$

Step 9.2.3

Rewrite the expression.

$\frac{q^{\frac{23}{5}}}{4^{2} {(p^{\frac{3}{2}})}^{2}}$

Step 10

Simplify the denominator.

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Step 10.1

Raise $4$ to the power of $2$ .

$\frac{q^{\frac{23}{5}}}{16 {(p^{\frac{3}{2}})}^{2}}$

Step 10.2

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

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Step 10.2.1

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

$\frac{q^{\frac{23}{5}}}{16 p^{\frac{3}{2} \cdot 2}}$

Step 10.2.2

Cancel the common factor of $2$ .

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Step 10.2.2.1

Cancel the common factor.

$\frac{q^{\frac{23}{5}}}{16 p^{\frac{3}{2} \cdot 2}}$

Step 10.2.2.2

Rewrite the expression.

$\frac{q^{\frac{23}{5}}}{16 p^{3}}$