Algebra Examples

${(3 m^{2} n^{2} - 3 p^{2})}^{3}$

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(3 m^{2} n^{2})}^{3} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4

Simplify each term.

Tap for more steps...

Step 4.1

Multiply ${(3 m^{2} n^{2})}^{3}$ by $1$ .

${(3 m^{2} n^{2})}^{3} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

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 $3 m^{2} n^{2}$ .

${(3 m^{2})}^{3} {(n^{2})}^{3} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.2.2

Apply the product rule to $3 m^{2}$ .

$3^{3} {(m^{2})}^{3} {(n^{2})}^{3} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.3

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

$27 {(m^{2})}^{3} {(n^{2})}^{3} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.4

Multiply the exponents in ${(m^{2})}^{3}$ .

Tap for more steps...

Step 4.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{2 \cdot 3} {(n^{2})}^{3} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.4.2

Multiply $2$ by $3$ .

$27 m^{6} {(n^{2})}^{3} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.5

Multiply the exponents in ${(n^{2})}^{3}$ .

Tap for more steps...

Step 4.5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{6} n^{2 \cdot 3} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.5.2

Multiply $2$ by $3$ .

$27 m^{6} n^{6} \cdot {(- 3 p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.6

Apply the product rule to $- 3 p^{2}$ .

$27 m^{6} n^{6} \cdot ({(- 3)}^{0} {(p^{2})}^{0}) + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.7

Anything raised to $0$ is $1$ .

$27 m^{6} n^{6} \cdot (1 {(p^{2})}^{0}) + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.8

Multiply ${(p^{2})}^{0}$ by $1$ .

$27 m^{6} n^{6} \cdot {(p^{2})}^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.9

Multiply the exponents in ${(p^{2})}^{0}$ .

Tap for more steps...

Step 4.9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{6} n^{6} \cdot p^{2 \cdot 0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.9.2

Multiply $2$ by $0$ .

$27 m^{6} n^{6} \cdot p^{0} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.10

Anything raised to $0$ is $1$ .

$27 m^{6} n^{6} \cdot 1 + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.11

Multiply $27$ by $1$ .

$27 m^{6} n^{6} + 3 \cdot {(3 m^{2} n^{2})}^{2} \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.12

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

Tap for more steps...

Step 4.12.1

Apply the product rule to $3 m^{2} n^{2}$ .

$27 m^{6} n^{6} + 3 \cdot ({(3 m^{2})}^{2} {(n^{2})}^{2}) \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.12.2

Apply the product rule to $3 m^{2}$ .

$27 m^{6} n^{6} + 3 \cdot (3^{2} {(m^{2})}^{2} {(n^{2})}^{2}) \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.13

Multiply $3$ by $3^{2}$ by adding the exponents.

Tap for more steps...

Step 4.13.1

Move $3^{2}$ .

$27 m^{6} n^{6} + 3^{2} \cdot 3 \cdot ({(m^{2})}^{2} {(n^{2})}^{2}) \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.13.2

Multiply $3^{2}$ by $3$ .

Tap for more steps...

Step 4.13.2.1

Raise $3$ to the power of $1$ .

$27 m^{6} n^{6} + 3^{2} \cdot 3^{1} \cdot ({(m^{2})}^{2} {(n^{2})}^{2}) \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.13.2.2

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

$27 m^{6} n^{6} + 3^{2 + 1} \cdot ({(m^{2})}^{2} {(n^{2})}^{2}) \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.13.3

Add $2$ and $1$ .

$27 m^{6} n^{6} + 3^{3} \cdot ({(m^{2})}^{2} {(n^{2})}^{2}) \cdot {(- 3 p^{2})}^{1} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.14

Simplify $3^{3} \cdot ({(m^{2})}^{2} {(n^{2})}^{2}) \cdot {(- 3 p^{2})}^{1}$ .

$27 m^{6} n^{6} + 3^{3} \cdot ({(m^{2})}^{2} {(n^{2})}^{2}) \cdot (- 3 p^{2}) + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.15

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

$27 m^{6} n^{6} + 27 \cdot ({(m^{2})}^{2} {(n^{2})}^{2}) \cdot (- 3 p^{2}) + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.16

Multiply the exponents in ${(m^{2})}^{2}$ .

Tap for more steps...

Step 4.16.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{6} n^{6} + 27 \cdot (m^{2 \cdot 2} {(n^{2})}^{2}) \cdot (- 3 p^{2}) + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.16.2

Multiply $2$ by $2$ .

$27 m^{6} n^{6} + 27 \cdot (m^{4} {(n^{2})}^{2}) \cdot (- 3 p^{2}) + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.17

Multiply the exponents in ${(n^{2})}^{2}$ .

Tap for more steps...

Step 4.17.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{6} n^{6} + 27 \cdot (m^{4} n^{2 \cdot 2}) \cdot (- 3 p^{2}) + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.17.2

Multiply $2$ by $2$ .

$27 m^{6} n^{6} + 27 \cdot (m^{4} n^{4}) \cdot (- 3 p^{2}) + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.18

Multiply $- 3$ by $27$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 3 \cdot {(3 m^{2} n^{2})}^{1} \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.19

Simplify.

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 3 \cdot (3 m^{2} n^{2}) \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.20

Multiply $3$ by $3$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 9 (m^{2} n^{2}) \cdot {(- 3 p^{2})}^{2} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.21

Apply the product rule to $- 3 p^{2}$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 9 m^{2} n^{2} \cdot ({(- 3)}^{2} {(p^{2})}^{2}) + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.22

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

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 9 m^{2} n^{2} \cdot (9 {(p^{2})}^{2}) + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.23

Multiply the exponents in ${(p^{2})}^{2}$ .

Tap for more steps...

Step 4.23.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 9 m^{2} n^{2} \cdot (9 p^{2 \cdot 2}) + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.23.2

Multiply $2$ by $2$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 9 m^{2} n^{2} \cdot (9 p^{4}) + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.24

Multiply $9$ by $9$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + 1 \cdot {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.25

Multiply ${(3 m^{2} n^{2})}^{0}$ by $1$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + {(3 m^{2} n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.26

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

Tap for more steps...

Step 4.26.1

Apply the product rule to $3 m^{2} n^{2}$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + {(3 m^{2})}^{0} {(n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.26.2

Apply the product rule to $3 m^{2}$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + 3^{0} {(m^{2})}^{0} {(n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.27

Anything raised to $0$ is $1$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + 1 {(m^{2})}^{0} {(n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.28

Multiply ${(m^{2})}^{0}$ by $1$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + {(m^{2})}^{0} {(n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.29

Multiply the exponents in ${(m^{2})}^{0}$ .

Tap for more steps...

Step 4.29.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + m^{2 \cdot 0} {(n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.29.2

Multiply $2$ by $0$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + m^{0} {(n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.30

Anything raised to $0$ is $1$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + 1 {(n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.31

Multiply ${(n^{2})}^{0}$ by $1$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + {(n^{2})}^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.32

Multiply the exponents in ${(n^{2})}^{0}$ .

Tap for more steps...

Step 4.32.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + n^{2 \cdot 0} \cdot {(- 3 p^{2})}^{3}$

Step 4.32.2

Multiply $2$ by $0$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + n^{0} \cdot {(- 3 p^{2})}^{3}$

Step 4.33

Anything raised to $0$ is $1$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + 1 \cdot {(- 3 p^{2})}^{3}$

Step 4.34

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

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + {(- 3 p^{2})}^{3}$

Step 4.35

Apply the product rule to $- 3 p^{2}$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} + {(- 3)}^{3} {(p^{2})}^{3}$

Step 4.36

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

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} - 27 {(p^{2})}^{3}$

Step 4.37

Multiply the exponents in ${(p^{2})}^{3}$ .

Tap for more steps...

Step 4.37.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} - 27 p^{2 \cdot 3}$

Step 4.37.2

Multiply $2$ by $3$ .

$27 m^{6} n^{6} - 81 m^{4} n^{4} p^{2} + 81 m^{2} n^{2} p^{4} - 27 p^{6}$