Algebra Examples

${(\frac{x^{\frac{1}{2}} y^{3}}{y^{\frac{3}{2}}})}^{3}$

Step 1

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

${(x^{\frac{1}{2}} y^{3} y^{- \frac{3}{2}})}^{3}$

Step 2

Multiply $y^{3}$ by $y^{- \frac{3}{2}}$ by adding the exponents.

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Step 2.1

Move $y^{- \frac{3}{2}}$ .

${(x^{\frac{1}{2}} (y^{- \frac{3}{2}} y^{3}))}^{3}$

Step 2.2

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

${(x^{\frac{1}{2}} y^{- \frac{3}{2} + 3})}^{3}$

Step 2.3

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

${(x^{\frac{1}{2}} y^{- \frac{3}{2} + 3 \cdot \frac{2}{2}})}^{3}$

Step 2.4

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

${(x^{\frac{1}{2}} y^{- \frac{3}{2} + \frac{3 \cdot 2}{2}})}^{3}$

Step 2.5

Combine the numerators over the common denominator.

${(x^{\frac{1}{2}} y^{\frac{- 3 + 3 \cdot 2}{2}})}^{3}$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Multiply $3$ by $2$ .

${(x^{\frac{1}{2}} y^{\frac{- 3 + 6}{2}})}^{3}$

Step 2.6.2

Add $- 3$ and $6$ .

${(x^{\frac{1}{2}} y^{\frac{3}{2}})}^{3}$

Step 3

Apply the product rule to $x^{\frac{1}{2}} y^{\frac{3}{2}}$ .

${(x^{\frac{1}{2}})}^{3} {(y^{\frac{3}{2}})}^{3}$

Step 4

Multiply the exponents in ${(x^{\frac{1}{2}})}^{3}$ .

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Step 4.1

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

$x^{\frac{1}{2} \cdot 3} {(y^{\frac{3}{2}})}^{3}$

Step 4.2

Combine $\frac{1}{2}$ and $3$ .

$x^{\frac{3}{2}} {(y^{\frac{3}{2}})}^{3}$

Step 5

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

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Step 5.1

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

$x^{\frac{3}{2}} y^{\frac{3}{2} \cdot 3}$

Step 5.2

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

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Step 5.2.1

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

$x^{\frac{3}{2}} y^{\frac{3 \cdot 3}{2}}$

Step 5.2.2

Multiply $3$ by $3$ .

$x^{\frac{3}{2}} y^{\frac{9}{2}}$