Algebra Examples

${(x^{\frac{1}{3}} \cdot y^{\frac{1}{2}})}^{9} \cdot \sqrt{y}$

Step 1

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

${(x^{\frac{1}{3}})}^{9} {(y^{\frac{1}{2}})}^{9} \cdot \sqrt{y}$

Step 2

Multiply the exponents in ${(x^{\frac{1}{3}})}^{9}$ .

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Step 2.1

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

$x^{\frac{1}{3} \cdot 9} {(y^{\frac{1}{2}})}^{9} \cdot \sqrt{y}$

Step 2.2

Cancel the common factor of $3$ .

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Step 2.2.1

Factor $3$ out of $9$ .

$x^{\frac{1}{3} \cdot (3 (3))} {(y^{\frac{1}{2}})}^{9} \cdot \sqrt{y}$

Step 2.2.2

Cancel the common factor.

$x^{\frac{1}{3} \cdot (3 \cdot 3)} {(y^{\frac{1}{2}})}^{9} \cdot \sqrt{y}$

Step 2.2.3

Rewrite the expression.

$x^{3} {(y^{\frac{1}{2}})}^{9} \cdot \sqrt{y}$

Step 3

Multiply the exponents in ${(y^{\frac{1}{2}})}^{9}$ .

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Step 3.1

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

$x^{3} y^{\frac{1}{2} \cdot 9} \cdot \sqrt{y}$

Step 3.2

Combine $\frac{1}{2}$ and $9$ .

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

Step 4

Multiply $x^{3} y^{\frac{9}{2}} \sqrt{y}$ .

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Step 4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$x^{3} (y^{\frac{1}{2}} y^{\frac{9}{2}})$

Step 4.2

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

$x^{3} y^{\frac{1}{2} + \frac{9}{2}}$

Step 4.3

Combine the numerators over the common denominator.

$x^{3} y^{\frac{1 + 9}{2}}$

Step 4.4

Add $1$ and $9$ .

$x^{3} y^{\frac{10}{2}}$

Step 4.5

Cancel the common factor of $10$ and $2$ .

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Step 4.5.1

Factor $2$ out of $10$ .

$x^{3} y^{\frac{2 \cdot 5}{2}}$

Step 4.5.2

Cancel the common factors.

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Step 4.5.2.1

Factor $2$ out of $2$ .

$x^{3} y^{\frac{2 \cdot 5}{2 (1)}}$

Step 4.5.2.2

Cancel the common factor.

$x^{3} y^{\frac{2 \cdot 5}{2 \cdot 1}}$

Step 4.5.2.3

Rewrite the expression.

$x^{3} y^{\frac{5}{1}}$

Step 4.5.2.4

Divide $5$ by $1$ .

$x^{3} y^{5}$