Algebra Examples

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

Step 1

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

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Step 1.1

Combine $a^{2}$ and $\frac{1}{2}$ .

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

Step 1.2

Combine $\frac{a^{2}}{2}$ and $b^{3}$ .

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Apply the product rule to $a^{2} b^{3}$ .

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

Step 3

Simplify the numerator.

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Step 3.1

Multiply the exponents in ${(a^{2})}^{3}$ .

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Step 3.1.1

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

$2 \frac{a^{2 \cdot 3} {(b^{3})}^{3}}{2^{3}}$

Step 3.1.2

Multiply $2$ by $3$ .

$2 \frac{a^{6} {(b^{3})}^{3}}{2^{3}}$

Step 3.2

Multiply the exponents in ${(b^{3})}^{3}$ .

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Step 3.2.1

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

$2 \frac{a^{6} b^{3 \cdot 3}}{2^{3}}$

Step 3.2.2

Multiply $3$ by $3$ .

$2 \frac{a^{6} b^{9}}{2^{3}}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Raise $2$ to the power of $3$ .

$2 \frac{a^{6} b^{9}}{8}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $8$ .

$2 \frac{a^{6} b^{9}}{2 (4)}$

Step 4.2.2

Cancel the common factor.

$2 \frac{a^{6} b^{9}}{2 \cdot 4}$

Step 4.2.3

Rewrite the expression.

$\frac{a^{6} b^{9}}{4}$