Basic Math Examples

$(\frac{5}{3} a^{3} - \frac{6}{4} b^{2}) (\frac{6}{4} b^{2} + \frac{5}{3} a)$

Step 1

Simplify terms.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Combine $\frac{5}{3}$ and $a^{3}$ .

$(\frac{5 a^{3}}{3} - \frac{6}{4} b^{2}) (\frac{6}{4} b^{2} + \frac{5}{3} a)$

Step 1.1.2

Cancel the common factor of $6$ and $4$ .

Tap for more steps...

Step 1.1.2.1

Factor $2$ out of $6$ .

$(\frac{5 a^{3}}{3} - \frac{2 (3)}{4} b^{2}) (\frac{6}{4} b^{2} + \frac{5}{3} a)$

Step 1.1.2.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.2.1

Factor $2$ out of $4$ .

$(\frac{5 a^{3}}{3} - \frac{2 \cdot 3}{2 \cdot 2} b^{2}) (\frac{6}{4} b^{2} + \frac{5}{3} a)$

Step 1.1.2.2.2

Cancel the common factor.

$(\frac{5 a^{3}}{3} - \frac{2 \cdot 3}{2 \cdot 2} b^{2}) (\frac{6}{4} b^{2} + \frac{5}{3} a)$

Step 1.1.2.2.3

Rewrite the expression.

$(\frac{5 a^{3}}{3} - \frac{3}{2} b^{2}) (\frac{6}{4} b^{2} + \frac{5}{3} a)$

Step 1.1.3

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

$(\frac{5 a^{3}}{3} - \frac{b^{2} \cdot 3}{2}) (\frac{6}{4} b^{2} + \frac{5}{3} a)$

Step 1.1.4

Move $3$ to the left of $b^{2}$ .

$(\frac{5 a^{3}}{3} - \frac{3 b^{2}}{2}) (\frac{6}{4} b^{2} + \frac{5}{3} a)$

Step 1.2

Simplify each term.

Tap for more steps...

Step 1.2.1

Cancel the common factor of $6$ and $4$ .

Tap for more steps...

Step 1.2.1.1

Factor $2$ out of $6$ .

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

Step 1.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.1.2.1

Factor $2$ out of $4$ .

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

Step 1.2.1.2.2

Cancel the common factor.

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

Step 1.2.1.2.3

Rewrite the expression.

$(\frac{5 a^{3}}{3} - \frac{3 b^{2}}{2}) (\frac{3}{2} b^{2} + \frac{5}{3} a)$

Step 1.2.2

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

$(\frac{5 a^{3}}{3} - \frac{3 b^{2}}{2}) (\frac{3 b^{2}}{2} + \frac{5}{3} a)$

Step 1.2.3

Combine $\frac{5}{3}$ and $a$ .

$(\frac{5 a^{3}}{3} - \frac{3 b^{2}}{2}) (\frac{3 b^{2}}{2} + \frac{5 a}{3})$

Step 2

Expand $(\frac{5 a^{3}}{3} - \frac{3 b^{2}}{2}) (\frac{3 b^{2}}{2} + \frac{5 a}{3})$ using the FOIL Method.

Tap for more steps...

Step 2.1

Apply the distributive property.

$\frac{5 a^{3}}{3} (\frac{3 b^{2}}{2} + \frac{5 a}{3}) - \frac{3 b^{2}}{2} (\frac{3 b^{2}}{2} + \frac{5 a}{3})$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

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

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Combine.

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

Step 3.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

Combine.

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

Step 3.4

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

Tap for more steps...

Step 3.4.1

Move $a$ .

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

Step 3.4.2

Multiply $a$ by $a^{3}$ .

Tap for more steps...

Step 3.4.2.1

Raise $a$ to the power of $1$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{5 (a^{1} a^{3}) \cdot 5}{3 \cdot 3} - \frac{3 b^{2}}{2} \cdot \frac{3 b^{2}}{2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.4.2.2

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

$\frac{5 a^{3} b^{2}}{2} + \frac{5 a^{1 + 3} \cdot 5}{3 \cdot 3} - \frac{3 b^{2}}{2} \cdot \frac{3 b^{2}}{2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.4.3

Add $1$ and $3$ .

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

Step 3.5

Multiply $5$ by $5$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{3 \cdot 3} - \frac{3 b^{2}}{2} \cdot \frac{3 b^{2}}{2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.6

Multiply $3$ by $3$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{3 b^{2}}{2} \cdot \frac{3 b^{2}}{2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.7

Multiply $- \frac{3 b^{2}}{2} \cdot \frac{3 b^{2}}{2}$ .

Tap for more steps...

Step 3.7.1

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

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{3 b^{2} (3 b^{2})}{2 \cdot 2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.7.2

Multiply $3$ by $3$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{2} b^{2}}{2 \cdot 2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.7.3

Multiply $b^{2}$ by $b^{2}$ by adding the exponents.

Tap for more steps...

Step 3.7.3.1

Move $b^{2}$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 (b^{2} b^{2})}{2 \cdot 2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.7.3.2

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

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{2 + 2}}{2 \cdot 2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.7.3.3

Add $2$ and $2$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{2 \cdot 2} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.7.4

Multiply $2$ by $2$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} - \frac{3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.8

Cancel the common factor of $3$ .

Tap for more steps...

Step 3.8.1

Move the leading negative in $- \frac{3 b^{2}}{2}$ into the numerator.

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} + \frac{- 3 b^{2}}{2} \cdot \frac{5 a}{3}$

Step 3.8.2

Factor $3$ out of $- 3 b^{2}$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} + \frac{3 (- b^{2})}{2} \cdot \frac{5 a}{3}$

Step 3.8.3

Cancel the common factor.

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} + \frac{3 (- b^{2})}{2} \cdot \frac{5 a}{3}$

Step 3.8.4

Rewrite the expression.

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} + \frac{- b^{2}}{2} (5 a)$

Step 3.9

Combine $5$ and $\frac{- b^{2}}{2}$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} + \frac{5 (- b^{2})}{2} a$

Step 3.10

Multiply $- 1$ by $5$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} + \frac{- 5 b^{2}}{2} a$

Step 3.11

Combine $\frac{- 5 b^{2}}{2}$ and $a$ .

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} + \frac{- 5 b^{2} a}{2}$

Step 3.12

Move the negative in front of the fraction.

$\frac{5 a^{3} b^{2}}{2} + \frac{25 a^{4}}{9} - \frac{9 b^{4}}{4} - \frac{5 b^{2} a}{2}$