Algebra Examples

${(\frac{a}{2})}^{4} \frac{{(8 a^{5})}^{2}}{a^{- 1} a^{10}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $8 a^{5}$ .

${(\frac{a}{2})}^{4} \frac{8^{2} {(a^{5})}^{2}}{a^{- 1} a^{10}}$

Step 1.2

Raise $8$ to the power of $2$ .

${(\frac{a}{2})}^{4} \frac{64 {(a^{5})}^{2}}{a^{- 1} a^{10}}$

Step 1.3

Multiply the exponents in ${(a^{5})}^{2}$ .

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Step 1.3.1

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

${(\frac{a}{2})}^{4} \frac{64 a^{5 \cdot 2}}{a^{- 1} a^{10}}$

Step 1.3.2

Multiply $5$ by $2$ .

${(\frac{a}{2})}^{4} \frac{64 a^{10}}{a^{- 1} a^{10}}$

Step 2

Multiply $a^{- 1}$ by $a^{10}$ by adding the exponents.

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Step 2.1

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

${(\frac{a}{2})}^{4} \frac{64 a^{10}}{a^{- 1 + 10}}$

Step 2.2

Add $- 1$ and $10$ .

${(\frac{a}{2})}^{4} \frac{64 a^{10}}{a^{9}}$

Step 3

Apply the product rule to $\frac{a}{2}$ .

$\frac{a^{4}}{2^{4}} \cdot \frac{64 a^{10}}{a^{9}}$

Step 4

Combine.

$\frac{a^{4} (64 a^{10})}{2^{4} a^{9}}$

Step 5

Multiply $a^{4}$ by $a^{10}$ by adding the exponents.

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Step 5.1

Move $a^{10}$ .

$\frac{a^{10} a^{4} \cdot 64}{2^{4} a^{9}}$

Step 5.2

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

$\frac{a^{10 + 4} \cdot 64}{2^{4} a^{9}}$

Step 5.3

Add $10$ and $4$ .

$\frac{a^{14} \cdot 64}{2^{4} a^{9}}$

Step 6

Cancel the common factor of $a^{14}$ and $a^{9}$ .

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Step 6.1

Factor $a^{9}$ out of $a^{14} \cdot 64$ .

$\frac{a^{9} (a^{5} \cdot 64)}{2^{4} a^{9}}$

Step 6.2

Cancel the common factors.

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Step 6.2.1

Factor $a^{9}$ out of $2^{4} a^{9}$ .

$\frac{a^{9} (a^{5} \cdot 64)}{a^{9} \cdot 2^{4}}$

Step 6.2.2

Cancel the common factor.

$\frac{a^{9} (a^{5} \cdot 64)}{a^{9} \cdot 2^{4}}$

Step 6.2.3

Rewrite the expression.

$\frac{a^{5} \cdot 64}{2^{4}}$

Step 7

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

$\frac{a^{5} \cdot 64}{16}$

Step 8

Cancel the common factor of $64$ and $16$ .

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Step 8.1

Factor $16$ out of $a^{5} \cdot 64$ .

$\frac{16 (a^{5} \cdot 4)}{16}$

Step 8.2

Cancel the common factors.

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Step 8.2.1

Factor $16$ out of $16$ .

$\frac{16 (a^{5} \cdot 4)}{16 (1)}$

Step 8.2.2

Cancel the common factor.

$\frac{16 (a^{5} \cdot 4)}{16 \cdot 1}$

Step 8.2.3

Rewrite the expression.

$\frac{a^{5} \cdot 4}{1}$

Step 8.2.4

Divide $a^{5} \cdot 4$ by $1$ .

$a^{5} \cdot 4$

Step 9

Move $4$ to the left of $a^{5}$ .

$4 a^{5}$