Basic Math Examples

${(\frac{b^{- \frac{5}{2}}}{c^{- \frac{2}{3}}})}^{2} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 1

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

${(\frac{1}{c^{- \frac{2}{3}} b^{\frac{5}{2}}})}^{2} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 2

Move $c^{- \frac{2}{3}}$ to the numerator using the negative exponent rule $\frac{1}{b^{- n}} = b^{n}$ .

${(\frac{c^{\frac{2}{3}}}{b^{\frac{5}{2}}})}^{2} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 3

Apply the product rule to $\frac{c^{\frac{2}{3}}}{b^{\frac{5}{2}}}$ .

$\frac{{(c^{\frac{2}{3}})}^{2}}{{(b^{\frac{5}{2}})}^{2}} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 4

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

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Step 4.1

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

$\frac{c^{\frac{2}{3} \cdot 2}}{{(b^{\frac{5}{2}})}^{2}} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 4.2

Multiply $\frac{2}{3} \cdot 2$ .

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Step 4.2.1

Combine $\frac{2}{3}$ and $2$ .

$\frac{c^{\frac{2 \cdot 2}{3}}}{{(b^{\frac{5}{2}})}^{2}} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 4.2.2

Multiply $2$ by $2$ .

$\frac{c^{\frac{4}{3}}}{{(b^{\frac{5}{2}})}^{2}} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 5

Multiply the exponents in ${(b^{\frac{5}{2}})}^{2}$ .

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Step 5.1

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

$\frac{c^{\frac{4}{3}}}{b^{\frac{5}{2} \cdot 2}} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Cancel the common factor.

$\frac{c^{\frac{4}{3}}}{b^{\frac{5}{2} \cdot 2}} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 5.2.2

Rewrite the expression.

$\frac{c^{\frac{4}{3}}}{b^{5}} {(b^{- \frac{1}{2}} c^{- \frac{1}{3}})}^{- 1}$

Step 6

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

$\frac{c^{\frac{4}{3}}}{b^{5}} {(\frac{1}{b^{\frac{1}{2}}} c^{- \frac{1}{3}})}^{- 1}$

Step 7

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

$\frac{c^{\frac{4}{3}}}{b^{5}} {(\frac{1}{b^{\frac{1}{2}}} \cdot \frac{1}{c^{\frac{1}{3}}})}^{- 1}$

Step 8

Combine.

$\frac{c^{\frac{4}{3}}}{b^{5}} {(\frac{1 \cdot 1}{b^{\frac{1}{2}} c^{\frac{1}{3}}})}^{- 1}$

Step 9

Multiply $1$ by $1$ .

$\frac{c^{\frac{4}{3}}}{b^{5}} {(\frac{1}{b^{\frac{1}{2}} c^{\frac{1}{3}}})}^{- 1}$

Step 10

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

$\frac{c^{\frac{4}{3}}}{b^{5}} (b^{\frac{1}{2}} c^{\frac{1}{3}})$

Step 11

Cancel the common factor of $b^{\frac{1}{2}}$ .

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Step 11.1

Factor $b^{\frac{1}{2}}$ out of $b^{5}$ .

$\frac{c^{\frac{4}{3}}}{b^{\frac{1}{2}} b^{\frac{9}{2}}} (b^{\frac{1}{2}} c^{\frac{1}{3}})$

Step 11.2

Factor $b^{\frac{1}{2}}$ out of $b^{\frac{1}{2}} c^{\frac{1}{3}}$ .

$\frac{c^{\frac{4}{3}}}{b^{\frac{1}{2}} b^{\frac{9}{2}}} (b^{\frac{1}{2}} (c^{\frac{1}{3}}))$

Step 11.3

Cancel the common factor.

$\frac{c^{\frac{4}{3}}}{b^{\frac{1}{2}} b^{\frac{9}{2}}} (b^{\frac{1}{2}} c^{\frac{1}{3}})$

Step 11.4

Rewrite the expression.

$\frac{c^{\frac{4}{3}}}{b^{\frac{9}{2}}} c^{\frac{1}{3}}$

Step 12

Combine $\frac{c^{\frac{4}{3}}}{b^{\frac{9}{2}}}$ and $c^{\frac{1}{3}}$ .

$\frac{c^{\frac{4}{3}} c^{\frac{1}{3}}}{b^{\frac{9}{2}}}$

Step 13

Multiply $c^{\frac{4}{3}}$ by $c^{\frac{1}{3}}$ by adding the exponents.

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Step 13.1

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

$\frac{c^{\frac{4}{3} + \frac{1}{3}}}{b^{\frac{9}{2}}}$

Step 13.2

Combine the numerators over the common denominator.

$\frac{c^{\frac{4 + 1}{3}}}{b^{\frac{9}{2}}}$

Step 13.3

Add $4$ and $1$ .

$\frac{c^{\frac{5}{3}}}{b^{\frac{9}{2}}}$