Algebra Examples

\frac{a^{\frac{5}{4}} {(2 a^{\frac{3}{4}})}^{3}}{a^{\frac{1}{4}}}

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

Step 1

Move

a^{\frac{1}{4}}

$a^{\frac{1}{4}}$ to the numerator using the negative exponent rule

\frac{1}{b^{n}} = b^{- n}

$\frac{1}{b^{n}} = b^{- n}$ .

a^{\frac{5}{4}} {(2 a^{\frac{3}{4}})}^{3} a^{- \frac{1}{4}}

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

Step 2

Multiply

a^{\frac{5}{4}}

$a^{\frac{5}{4}}$ by

a^{- \frac{1}{4}}

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

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Step 2.1

Move

a^{- \frac{1}{4}}

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

a^{- \frac{1}{4}} a^{\frac{5}{4}} {(2 a^{\frac{3}{4}})}^{3}

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

Step 2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

a^{- \frac{1}{4} + \frac{5}{4}} {(2 a^{\frac{3}{4}})}^{3}

$a^{- \frac{1}{4} + \frac{5}{4}} {(2 a^{\frac{3}{4}})}^{3}$

Step 2.3

Combine the numerators over the common denominator.

a^{\frac{- 1 + 5}{4}} {(2 a^{\frac{3}{4}})}^{3}

$a^{\frac{- 1 + 5}{4}} {(2 a^{\frac{3}{4}})}^{3}$

Step 2.4

Add

- 1

$- 1$ and

5

$5$ .

a^{\frac{4}{4}} {(2 a^{\frac{3}{4}})}^{3}

$a^{\frac{4}{4}} {(2 a^{\frac{3}{4}})}^{3}$

Step 2.5

Divide

4

$4$ by

4

$4$ .

a^{1} {(2 a^{\frac{3}{4}})}^{3}

$a^{1} {(2 a^{\frac{3}{4}})}^{3}$

a^{1} {(2 a^{\frac{3}{4}})}^{3}

$a^{1} {(2 a^{\frac{3}{4}})}^{3}$

Step 3

Simplify

a^{1} {(2 a^{\frac{3}{4}})}^{3}

$a^{1} {(2 a^{\frac{3}{4}})}^{3}$ .

a {(2 a^{\frac{3}{4}})}^{3}

$a {(2 a^{\frac{3}{4}})}^{3}$

Step 4

Apply the product rule to

2 a^{\frac{3}{4}}

$2 a^{\frac{3}{4}}$ .

a (2^{3} {(a^{\frac{3}{4}})}^{3})

$a (2^{3} {(a^{\frac{3}{4}})}^{3})$

Step 5

Rewrite using the commutative property of multiplication.

2^{3} a {(a^{\frac{3}{4}})}^{3}

$2^{3} a {(a^{\frac{3}{4}})}^{3}$

Step 6

Raise

2

$2$ to the power of

3

$3$ .

8 a {(a^{\frac{3}{4}})}^{3}

$8 a {(a^{\frac{3}{4}})}^{3}$

Step 7

Multiply the exponents in

{(a^{\frac{3}{4}})}^{3}

${(a^{\frac{3}{4}})}^{3}$ .

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Step 7.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

8 a \cdot a^{\frac{3}{4} \cdot 3}

$8 a \cdot a^{\frac{3}{4} \cdot 3}$

Step 7.2

Multiply

\frac{3}{4} \cdot 3

$\frac{3}{4} \cdot 3$ .

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Step 7.2.1

Combine

\frac{3}{4}

$\frac{3}{4}$ and

3

$3$ .

8 a \cdot a^{\frac{3 \cdot 3}{4}}

$8 a \cdot a^{\frac{3 \cdot 3}{4}}$

Step 7.2.2

Multiply

3

$3$ by

3

$3$ .

8 a \cdot a^{\frac{9}{4}}

$8 a \cdot a^{\frac{9}{4}}$

8 a \cdot a^{\frac{9}{4}}

$8 a \cdot a^{\frac{9}{4}}$

8 a \cdot a^{\frac{9}{4}}

$8 a \cdot a^{\frac{9}{4}}$

Step 8

Multiply

a

$a$ by

a^{\frac{9}{4}}

$a^{\frac{9}{4}}$ by adding the exponents.

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Step 8.1

Move

a^{\frac{9}{4}}

$a^{\frac{9}{4}}$ .

8 (a^{\frac{9}{4}} a)

$8 (a^{\frac{9}{4}} a)$

Step 8.2

Multiply

a^{\frac{9}{4}}

$a^{\frac{9}{4}}$ by

a

$a$ .

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Step 8.2.1

Raise

a

$a$ to the power of

1

$1$ .

8 (a^{\frac{9}{4}} a^{1})

$8 (a^{\frac{9}{4}} a^{1})$

Step 8.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

8 a^{\frac{9}{4} + 1}

$8 a^{\frac{9}{4} + 1}$

8 a^{\frac{9}{4} + 1}

$8 a^{\frac{9}{4} + 1}$

Step 8.3

Write

1

$1$ as a fraction with a common denominator.

8 a^{\frac{9}{4} + \frac{4}{4}}

$8 a^{\frac{9}{4} + \frac{4}{4}}$

Step 8.4

Combine the numerators over the common denominator.

8 a^{\frac{9 + 4}{4}}

$8 a^{\frac{9 + 4}{4}}$

Step 8.5

Add

9

$9$ and

4

$4$ .

8 a^{\frac{13}{4}}

$8 a^{\frac{13}{4}}$

8 a^{\frac{13}{4}}

$8 a^{\frac{13}{4}}$