Algebra Examples

$(2 a^{\frac{1}{3}} \cdot 3 b^{\frac{1}{2}}) (a^{\frac{1}{3}} \cdot 3 b^{\frac{3}{4}})$

Step 1

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

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Step 1.1

Move $a^{\frac{1}{3}}$ .

$2 (a^{\frac{1}{3}} a^{\frac{1}{3}}) \cdot (3 b^{\frac{1}{2}}) (3 b^{\frac{3}{4}})$

Step 1.2

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

$2 a^{\frac{1}{3} + \frac{1}{3}} \cdot (3 b^{\frac{1}{2}}) (3 b^{\frac{3}{4}})$

Step 1.3

Combine the numerators over the common denominator.

$2 a^{\frac{1 + 1}{3}} \cdot (3 b^{\frac{1}{2}}) (3 b^{\frac{3}{4}})$

Step 1.4

Add $1$ and $1$ .

$2 a^{\frac{2}{3}} \cdot (3 b^{\frac{1}{2}}) (3 b^{\frac{3}{4}})$

Step 2

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

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Step 2.1

Move $b^{\frac{3}{4}}$ .

$2 a^{\frac{2}{3}} \cdot (3 (b^{\frac{3}{4}} b^{\frac{1}{2}})) \cdot 3$

Step 2.2

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

$2 a^{\frac{2}{3}} \cdot (3 b^{\frac{3}{4} + \frac{1}{2}}) \cdot 3$

Step 2.3

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

$2 a^{\frac{2}{3}} \cdot (3 b^{\frac{3}{4} + \frac{1}{2} \cdot \frac{2}{2}}) \cdot 3$

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 $\frac{2}{2}$ .

$2 a^{\frac{2}{3}} \cdot (3 b^{\frac{3}{4} + \frac{2}{2 \cdot 2}}) \cdot 3$

Step 2.4.2

Multiply $2$ by $2$ .

$2 a^{\frac{2}{3}} \cdot (3 b^{\frac{3}{4} + \frac{2}{4}}) \cdot 3$

Step 2.5

Combine the numerators over the common denominator.

$2 a^{\frac{2}{3}} \cdot (3 b^{\frac{3 + 2}{4}}) \cdot 3$

Step 2.6

Add $3$ and $2$ .

$2 a^{\frac{2}{3}} \cdot (3 b^{\frac{5}{4}}) \cdot 3$

Step 3

Rewrite using the commutative property of multiplication.

$2 \cdot 3 a^{\frac{2}{3}} b^{\frac{5}{4}} \cdot 3$

Step 4

Multiply $2$ by $3$ .

$6 a^{\frac{2}{3}} b^{\frac{5}{4}} \cdot 3$

Step 5

Multiply $3$ by $6$ .

$18 a^{\frac{2}{3}} b^{\frac{5}{4}}$