Algebra Examples

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

Step 1

Multiply $x^{\frac{1}{2}}$ by $x^{\frac{2}{3}}$ by adding the exponents.

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Step 1.1

Move $x^{\frac{2}{3}}$ .

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 1.5

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.5.1

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

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

Step 1.5.2

Multiply $3$ by $2$ .

$2 x^{\frac{2 \cdot 2}{6} + \frac{1}{2} \cdot \frac{3}{3}} y^{\frac{3}{4}} (- 3 y^{\frac{1}{2}})$

Step 1.5.3

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

$2 x^{\frac{2 \cdot 2}{6} + \frac{3}{2 \cdot 3}} y^{\frac{3}{4}} (- 3 y^{\frac{1}{2}})$

Step 1.5.4

Multiply $2$ by $3$ .

$2 x^{\frac{2 \cdot 2}{6} + \frac{3}{6}} y^{\frac{3}{4}} (- 3 y^{\frac{1}{2}})$

Step 1.6

Combine the numerators over the common denominator.

$2 x^{\frac{2 \cdot 2 + 3}{6}} y^{\frac{3}{4}} (- 3 y^{\frac{1}{2}})$

Step 1.7

Simplify the numerator.

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Step 1.7.1

Multiply $2$ by $2$ .

$2 x^{\frac{4 + 3}{6}} y^{\frac{3}{4}} (- 3 y^{\frac{1}{2}})$

Step 1.7.2

Add $4$ and $3$ .

$2 x^{\frac{7}{6}} y^{\frac{3}{4}} (- 3 y^{\frac{1}{2}})$

Step 2

Multiply $y^{\frac{3}{4}}$ by $y^{\frac{1}{2}}$ by adding the exponents.

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Step 2.1

Move $y^{\frac{1}{2}}$ .

$2 x^{\frac{7}{6}} (y^{\frac{1}{2}} y^{\frac{3}{4}}) \cdot - 3$

Step 2.2

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

$2 x^{\frac{7}{6}} y^{\frac{1}{2} + \frac{3}{4}} \cdot - 3$

Step 2.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 x^{\frac{7}{6}} y^{\frac{1}{2} \cdot \frac{2}{2} + \frac{3}{4}} \cdot - 3$

Step 2.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$2 x^{\frac{7}{6}} y^{\frac{2}{2 \cdot 2} + \frac{3}{4}} \cdot - 3$

Step 2.4.2

Multiply $2$ by $2$ .

$2 x^{\frac{7}{6}} y^{\frac{2}{4} + \frac{3}{4}} \cdot - 3$

Step 2.5

Combine the numerators over the common denominator.

$2 x^{\frac{7}{6}} y^{\frac{2 + 3}{4}} \cdot - 3$

Step 2.6

Add $2$ and $3$ .

$2 x^{\frac{7}{6}} y^{\frac{5}{4}} \cdot - 3$

Step 3

Multiply $- 3$ by $2$ .

$- 6 x^{\frac{7}{6}} y^{\frac{5}{4}}$