Algebra Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(a)}^{2 - k} \cdot {(- \frac{3}{2})}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} {(a)}^{2 - 0} \cdot {(- \frac{3}{2})}^{0} + \frac{2!}{(2 - 1)! 1!} {(a)}^{2 - 1} \cdot {(- \frac{3}{2})}^{1} + \frac{2!}{(2 - 2)! 2!} {(a)}^{2 - 2} \cdot {(- \frac{3}{2})}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(a)}^{2} \cdot {(- \frac{3}{2})}^{0} + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(a)}^{2}$ by $1$ .

${(a)}^{2} \cdot {(- \frac{3}{2})}^{0} + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.2.1

Apply the product rule to $- \frac{3}{2}$ .

$a^{2} \cdot ({(- 1)}^{0} {(\frac{3}{2})}^{0}) + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.2.2

Apply the product rule to $\frac{3}{2}$ .

$a^{2} \cdot ({(- 1)}^{0} \frac{3^{0}}{2^{0}}) + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.3

Anything raised to $0$ is $1$ .

$a^{2} \cdot (1 \frac{3^{0}}{2^{0}}) + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.4

Multiply $\frac{3^{0}}{2^{0}}$ by $1$ .

$a^{2} \cdot \frac{3^{0}}{2^{0}} + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.5

Anything raised to $0$ is $1$ .

$a^{2} \cdot \frac{1}{2^{0}} + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.6

Anything raised to $0$ is $1$ .

$a^{2} \cdot \frac{1}{1} + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.7

Divide $1$ by $1$ .

$a^{2} \cdot 1 + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.8

Multiply $a^{2}$ by $1$ .

$a^{2} + 2 \cdot {(a)}^{1} \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.9

Simplify.

$a^{2} + 2 \cdot a \cdot {(- \frac{3}{2})}^{1} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.10

Simplify.

$a^{2} + 2 a \cdot (- \frac{3}{2}) + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.11

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.11.1

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

$a^{2} + 2 a \cdot \frac{- 3}{2} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.11.2

Factor $2$ out of $2 a$ .

$a^{2} + 2 (a) \cdot \frac{- 3}{2} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.11.3

Cancel the common factor.

$a^{2} + 2 a \cdot \frac{- 3}{2} + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.11.4

Rewrite the expression.

$a^{2} + a \cdot - 3 + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.12

Move $- 3$ to the left of $a$ .

$a^{2} - 3 \cdot a + 1 \cdot {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.13

Multiply ${(a)}^{0}$ by $1$ .

$a^{2} - 3 a + {(a)}^{0} \cdot {(- \frac{3}{2})}^{2}$

Step 4.14

Anything raised to $0$ is $1$ .

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

Step 4.15

Multiply ${(- \frac{3}{2})}^{2}$ by $1$ .

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

Step 4.16

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Tap for more steps...

Step 4.16.1

Apply the product rule to $- \frac{3}{2}$ .

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

Step 4.16.2

Apply the product rule to $\frac{3}{2}$ .

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

Step 4.17

Raise $- 1$ to the power of $2$ .

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

Step 4.18

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

$a^{2} - 3 a + \frac{3^{2}}{2^{2}}$

Step 4.19

Raise $3$ to the power of $2$ .

$a^{2} - 3 a + \frac{9}{2^{2}}$

Step 4.20

Raise $2$ to the power of $2$ .

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