Pre-Algebra Examples

$\frac{9^{\frac{5}{2}} a^{\frac{3}{4}} b^{\frac{5}{4}}}{{(81)}^{\frac{1}{2}} a^{- \frac{1}{4}} b^{- \frac{1}{2}}}$

Step 1

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

$\frac{9^{\frac{5}{2}} a^{\frac{3}{4}} b^{\frac{5}{4}} a^{\frac{1}{4}}}{{(81)}^{\frac{1}{2}} b^{- \frac{1}{2}}}$

Step 2

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

$\frac{9^{\frac{5}{2}} a^{\frac{3}{4}} b^{\frac{5}{4}} a^{\frac{1}{4}} b^{\frac{1}{2}}}{{(81)}^{\frac{1}{2}}}$

Step 3

Simplify the numerator.

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Step 3.1

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

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Step 3.1.1

Move $a^{\frac{1}{4}}$ .

$\frac{9^{\frac{5}{2}} (a^{\frac{1}{4}} a^{\frac{3}{4}}) b^{\frac{5}{4}} b^{\frac{1}{2}}}{{(81)}^{\frac{1}{2}}}$

Step 3.1.2

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

$\frac{9^{\frac{5}{2}} a^{\frac{1}{4} + \frac{3}{4}} b^{\frac{5}{4}} b^{\frac{1}{2}}}{{(81)}^{\frac{1}{2}}}$

Step 3.1.3

Combine the numerators over the common denominator.

$\frac{9^{\frac{5}{2}} a^{\frac{1 + 3}{4}} b^{\frac{5}{4}} b^{\frac{1}{2}}}{{(81)}^{\frac{1}{2}}}$

Step 3.1.4

Add $1$ and $3$ .

$\frac{9^{\frac{5}{2}} a^{\frac{4}{4}} b^{\frac{5}{4}} b^{\frac{1}{2}}}{{(81)}^{\frac{1}{2}}}$

Step 3.1.5

Divide $4$ by $4$ .

$\frac{9^{\frac{5}{2}} a^{1} b^{\frac{5}{4}} b^{\frac{1}{2}}}{{(81)}^{\frac{1}{2}}}$

Step 3.2

Simplify $9^{\frac{5}{2}} a^{1} b^{\frac{5}{4}}$ .

$\frac{9^{\frac{5}{2}} a b^{\frac{5}{4}} b^{\frac{1}{2}}}{{(81)}^{\frac{1}{2}}}$

Step 3.3

Multiply $b^{\frac{5}{4}}$ by $b^{\frac{1}{2}}$ by adding the exponents.

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Step 3.3.1

Move $b^{\frac{1}{2}}$ .

$\frac{9^{\frac{5}{2}} a (b^{\frac{1}{2}} b^{\frac{5}{4}})}{{(81)}^{\frac{1}{2}}}$

Step 3.3.2

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

$\frac{9^{\frac{5}{2}} a b^{\frac{1}{2} + \frac{5}{4}}}{{(81)}^{\frac{1}{2}}}$

Step 3.3.3

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

$\frac{9^{\frac{5}{2}} a b^{\frac{1}{2} \cdot \frac{2}{2} + \frac{5}{4}}}{{(81)}^{\frac{1}{2}}}$

Step 3.3.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.3.4.1

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

$\frac{9^{\frac{5}{2}} a b^{\frac{2}{2 \cdot 2} + \frac{5}{4}}}{{(81)}^{\frac{1}{2}}}$

Step 3.3.4.2

Multiply $2$ by $2$ .

$\frac{9^{\frac{5}{2}} a b^{\frac{2}{4} + \frac{5}{4}}}{{(81)}^{\frac{1}{2}}}$

Step 3.3.5

Combine the numerators over the common denominator.

$\frac{9^{\frac{5}{2}} a b^{\frac{2 + 5}{4}}}{{(81)}^{\frac{1}{2}}}$

Step 3.3.6

Add $2$ and $5$ .

$\frac{9^{\frac{5}{2}} a b^{\frac{7}{4}}}{{(81)}^{\frac{1}{2}}}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $9$ as $3^{2}$ .

$\frac{{(3^{2})}^{\frac{5}{2}} a b^{\frac{7}{4}}}{81^{\frac{1}{2}}}$

Step 4.2

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

$\frac{3^{2 (\frac{5}{2})} a b^{\frac{7}{4}}}{81^{\frac{1}{2}}}$

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

$\frac{3^{2 (\frac{5}{2})} a b^{\frac{7}{4}}}{81^{\frac{1}{2}}}$

Step 4.3.2

Rewrite the expression.

$\frac{3^{5} a b^{\frac{7}{4}}}{81^{\frac{1}{2}}}$

Step 4.4

Raise $3$ to the power of $5$ .

$\frac{243 a b^{\frac{7}{4}}}{81^{\frac{1}{2}}}$

Step 5

Simplify the denominator.

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Step 5.1

Rewrite $81$ as $9^{2}$ .

$\frac{243 a b^{\frac{7}{4}}}{{(9^{2})}^{\frac{1}{2}}}$

Step 5.2

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

$\frac{243 a b^{\frac{7}{4}}}{9^{2 (\frac{1}{2})}}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Cancel the common factor.

$\frac{243 a b^{\frac{7}{4}}}{9^{2 (\frac{1}{2})}}$

Step 5.3.2

Rewrite the expression.

$\frac{243 a b^{\frac{7}{4}}}{9^{1}}$

Step 5.4

Evaluate the exponent.

$\frac{243 a b^{\frac{7}{4}}}{9}$

Step 6

Factor $9$ out of $243 a b^{\frac{7}{4}}$ .

$\frac{9 (27 a b^{\frac{7}{4}})}{9}$

Step 7

Cancel the common factors.

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Step 7.1

Factor $9$ out of $9$ .

$\frac{9 (27 a b^{\frac{7}{4}})}{9 (1)}$

Step 7.2

Cancel the common factor.

$\frac{9 (27 a b^{\frac{7}{4}})}{9 \cdot 1}$

Step 7.3

Rewrite the expression.

$\frac{27 a b^{\frac{7}{4}}}{1}$

Step 7.4

Divide $27 a b^{\frac{7}{4}}$ by $1$ .

$27 a b^{\frac{7}{4}}$