Algebra Examples

${(9 n - \frac{4}{7} z)}^{2}$

Step 1

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

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(9 n)}^{2 - k} \cdot {(- \frac{4}{7} z)}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(9 n)}^{2 - 0} \cdot {(- \frac{4}{7} z)}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(9 n)}^{2 - 1} \cdot {(- \frac{4}{7} z)}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(9 n)}^{2 - 2} \cdot {(- \frac{4}{7} z)}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(9 n)}^{2} \cdot {(- \frac{4}{7} z)}^{0} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(9 n)}^{2}$ by $1$ .

${(9 n)}^{2} \cdot {(- \frac{4}{7} z)}^{0} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.2

Apply the product rule to $9 n$ .

$9^{2} n^{2} \cdot {(- \frac{4}{7} z)}^{0} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.3

Raise $9$ to the power of $2$ .

$81 n^{2} \cdot {(- \frac{4}{7} z)}^{0} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.4

Combine $z$ and $\frac{4}{7}$ .

$81 n^{2} \cdot {(- \frac{z \cdot 4}{7})}^{0} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.5

Move $4$ to the left of $z$ .

$81 n^{2} \cdot {(- \frac{4 \cdot z}{7})}^{0} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.6

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

Tap for more steps...

Step 4.6.1

Apply the product rule to $- \frac{4 z}{7}$ .

$81 n^{2} \cdot ({(- 1)}^{0} {(\frac{4 z}{7})}^{0}) + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.6.2

Apply the product rule to $\frac{4 z}{7}$ .

$81 n^{2} \cdot ({(- 1)}^{0} \frac{{(4 z)}^{0}}{7^{0}}) + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.6.3

Apply the product rule to $4 z$ .

$81 n^{2} \cdot ({(- 1)}^{0} \frac{4^{0} z^{0}}{7^{0}}) + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.7

Anything raised to $0$ is $1$ .

$81 n^{2} \cdot (1 \frac{4^{0} z^{0}}{7^{0}}) + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.8

Multiply $\frac{4^{0} z^{0}}{7^{0}}$ by $1$ .

$81 n^{2} \cdot \frac{4^{0} z^{0}}{7^{0}} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.9

Simplify the numerator.

Tap for more steps...

Step 4.9.1

Anything raised to $0$ is $1$ .

$81 n^{2} \cdot \frac{1 z^{0}}{7^{0}} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.9.2

Anything raised to $0$ is $1$ .

$81 n^{2} \cdot \frac{1 \cdot 1}{7^{0}} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.9.3

Multiply $1$ by $1$ .

$81 n^{2} \cdot \frac{1}{7^{0}} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.10

Anything raised to $0$ is $1$ .

$81 n^{2} \cdot \frac{1}{1} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.11

Divide $1$ by $1$ .

$81 n^{2} \cdot 1 + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.12

Multiply $81$ by $1$ .

$81 n^{2} + 2 \cdot {(9 n)}^{1} \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.13

Simplify.

$81 n^{2} + 2 \cdot (9 n) \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.14

Multiply $9$ by $2$ .

$81 n^{2} + 18 n \cdot {(- \frac{4}{7} z)}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.15

Combine $z$ and $\frac{4}{7}$ .

$81 n^{2} + 18 n \cdot {(- \frac{z \cdot 4}{7})}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.16

Move $4$ to the left of $z$ .

$81 n^{2} + 18 n \cdot {(- \frac{4 \cdot z}{7})}^{1} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.17

Multiply $18 n (- \frac{4 z}{7})$ .

Tap for more steps...

Step 4.17.1

Multiply $- 1$ by $18$ .

$81 n^{2} - 18 n \frac{4 z}{7} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.17.2

Combine $- 18$ and $\frac{4 z}{7}$ .

$81 n^{2} + n \frac{- 18 (4 z)}{7} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.17.3

Multiply $4$ by $- 18$ .

$81 n^{2} + n \frac{- 72 z}{7} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.17.4

Combine $n$ and $\frac{- 72 z}{7}$ .

$81 n^{2} + \frac{n (- 72 z)}{7} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.18

Remove unnecessary parentheses.

$81 n^{2} + \frac{n \cdot - 72 z}{7} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.19

Move $- 72$ to the left of $n$ .

$81 n^{2} + \frac{- 72 \cdot n z}{7} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.20

Move the negative in front of the fraction.

$81 n^{2} - \frac{72 \cdot n z}{7} + 1 \cdot {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.21

Multiply ${(9 n)}^{0}$ by $1$ .

$81 n^{2} - \frac{72 n z}{7} + {(9 n)}^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.22

Apply the product rule to $9 n$ .

$81 n^{2} - \frac{72 n z}{7} + 9^{0} n^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.23

Anything raised to $0$ is $1$ .

$81 n^{2} - \frac{72 n z}{7} + 1 n^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.24

Multiply $n^{0}$ by $1$ .

$81 n^{2} - \frac{72 n z}{7} + n^{0} \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.25

Anything raised to $0$ is $1$ .

$81 n^{2} - \frac{72 n z}{7} + 1 \cdot {(- \frac{4}{7} z)}^{2}$

Step 4.26

Multiply ${(- \frac{4}{7} z)}^{2}$ by $1$ .

$81 n^{2} - \frac{72 n z}{7} + {(- \frac{4}{7} z)}^{2}$

Step 4.27

Combine $z$ and $\frac{4}{7}$ .

$81 n^{2} - \frac{72 n z}{7} + {(- \frac{z \cdot 4}{7})}^{2}$

Step 4.28

Move $4$ to the left of $z$ .

$81 n^{2} - \frac{72 n z}{7} + {(- \frac{4 \cdot z}{7})}^{2}$

Step 4.29

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

Tap for more steps...

Step 4.29.1

Apply the product rule to $- \frac{4 z}{7}$ .

$81 n^{2} - \frac{72 n z}{7} + {(- 1)}^{2} {(\frac{4 z}{7})}^{2}$

Step 4.29.2

Apply the product rule to $\frac{4 z}{7}$ .

$81 n^{2} - \frac{72 n z}{7} + {(- 1)}^{2} \frac{{(4 z)}^{2}}{7^{2}}$

Step 4.29.3

Apply the product rule to $4 z$ .

$81 n^{2} - \frac{72 n z}{7} + {(- 1)}^{2} \frac{4^{2} z^{2}}{7^{2}}$

Step 4.30

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

$81 n^{2} - \frac{72 n z}{7} + 1 \frac{4^{2} z^{2}}{7^{2}}$

Step 4.31

Multiply $\frac{4^{2} z^{2}}{7^{2}}$ by $1$ .

$81 n^{2} - \frac{72 n z}{7} + \frac{4^{2} z^{2}}{7^{2}}$

Step 4.32

Raise $4$ to the power of $2$ .

$81 n^{2} - \frac{72 n z}{7} + \frac{16 z^{2}}{7^{2}}$

Step 4.33

Raise $7$ to the power of $2$ .

$81 n^{2} - \frac{72 n z}{7} + \frac{16 z^{2}}{49}$