Algebra Examples

${(x^{\frac{1}{2}} \cdot x^{2} y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 1

Multiply $x^{\frac{1}{2}}$ by $x^{2}$ by adding the exponents.

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Step 1.1

Move $x^{2}$ .

${(x^{2} x^{\frac{1}{2}} \cdot y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 1.2

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

${(x^{2 + \frac{1}{2}} \cdot y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 1.3

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

${(x^{2 \cdot \frac{2}{2} + \frac{1}{2}} \cdot y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 1.4

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

${(x^{\frac{2 \cdot 2}{2} + \frac{1}{2}} \cdot y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 1.5

Combine the numerators over the common denominator.

${(x^{\frac{2 \cdot 2 + 1}{2}} \cdot y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $2$ by $2$ .

${(x^{\frac{4 + 1}{2}} \cdot y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 1.6.2

Add $4$ and $1$ .

${(x^{\frac{5}{2}} \cdot y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 2

Apply the product rule to $x^{\frac{5}{2}} y^{\frac{4}{3}}$ .

${(x^{\frac{5}{2}})}^{\frac{3}{2}} {(y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 3

Multiply the exponents in ${(x^{\frac{5}{2}})}^{\frac{3}{2}}$ .

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Step 3.1

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

$x^{\frac{5}{2} \cdot \frac{3}{2}} {(y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 3.2

Multiply $\frac{5}{2} \cdot \frac{3}{2}$ .

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Step 3.2.1

Multiply $\frac{5}{2}$ by $\frac{3}{2}$ .

$x^{\frac{5 \cdot 3}{2 \cdot 2}} {(y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 3.2.2

Multiply $5$ by $3$ .

$x^{\frac{15}{2 \cdot 2}} {(y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 3.2.3

Multiply $2$ by $2$ .

$x^{\frac{15}{4}} {(y^{\frac{4}{3}})}^{\frac{3}{2}}$

Step 4

Multiply the exponents in ${(y^{\frac{4}{3}})}^{\frac{3}{2}}$ .

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Step 4.1

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

$x^{\frac{15}{4}} y^{\frac{4}{3} \cdot \frac{3}{2}}$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Factor $2$ out of $4$ .

$x^{\frac{15}{4}} y^{\frac{2 (2)}{3} \cdot \frac{3}{2}}$

Step 4.2.2

Cancel the common factor.

$x^{\frac{15}{4}} y^{\frac{2 \cdot 2}{3} \cdot \frac{3}{2}}$

Step 4.2.3

Rewrite the expression.

$x^{\frac{15}{4}} y^{\frac{2}{3} \cdot 3}$

Step 4.3

Cancel the common factor of $3$ .

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Step 4.3.1

Cancel the common factor.

$x^{\frac{15}{4}} y^{\frac{2}{3} \cdot 3}$

Step 4.3.2

Rewrite the expression.

$x^{\frac{15}{4}} y^{2}$

Step 5

Reorder factors in $x^{\frac{15}{4}} y^{2}$ .

$y^{2} x^{\frac{15}{4}}$