Precalculus Examples

${(2 n^{3} + 4 m^{2})}^{2}$

Step 1

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

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(2 n^{3})}^{2 - k} \cdot {(4 m^{2})}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(2 n^{3})}^{2 - 0} \cdot {(4 m^{2})}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(2 n^{3})}^{2 - 1} \cdot {(4 m^{2})}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(2 n^{3})}^{2 - 2} \cdot {(4 m^{2})}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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Step 4.4.1

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

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

Step 4.4.2

Multiply $3$ by $2$ .

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

Step 4.5

Apply the product rule to $4 m^{2}$ .

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

Step 4.6

Rewrite using the commutative property of multiplication.

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

Step 4.7

Multiply $4$ by $4^{0}$ by adding the exponents.

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Step 4.7.1

Multiply $4$ by $4^{0}$ .

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Step 4.7.1.1

Raise $4$ to the power of $1$ .

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

Step 4.7.1.2

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

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

Step 4.7.2

Add $1$ and $0$ .

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

Step 4.8

Simplify $4^{1} n^{6} {(m^{2})}^{0}$ .

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

Step 4.9

Simplify.

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

Step 4.10

Multiply $2$ by $2$ .

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

Step 4.11

Simplify.

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

Step 4.12

Rewrite using the commutative property of multiplication.

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

Step 4.13

Multiply $4$ by $4$ .

$4 n^{6} + 16 n^{3} m^{2} + 1 \cdot {(2 n^{3})}^{0} \cdot {(4 m^{2})}^{2}$

Step 4.14

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

$4 n^{6} + 16 n^{3} m^{2} + {(2 n^{3})}^{0} \cdot {(4 m^{2})}^{2}$

Step 4.15

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

$4 n^{6} + 16 n^{3} m^{2} + 2^{0} {(n^{3})}^{0} \cdot {(4 m^{2})}^{2}$

Step 4.16

Anything raised to $0$ is $1$ .

$4 n^{6} + 16 n^{3} m^{2} + 1 {(n^{3})}^{0} \cdot {(4 m^{2})}^{2}$

Step 4.17

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

$4 n^{6} + 16 n^{3} m^{2} + {(n^{3})}^{0} \cdot {(4 m^{2})}^{2}$

Step 4.18

Multiply the exponents in ${(n^{3})}^{0}$ .

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Step 4.18.1

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

$4 n^{6} + 16 n^{3} m^{2} + n^{3 \cdot 0} \cdot {(4 m^{2})}^{2}$

Step 4.18.2

Multiply $3$ by $0$ .

$4 n^{6} + 16 n^{3} m^{2} + n^{0} \cdot {(4 m^{2})}^{2}$

Step 4.19

Anything raised to $0$ is $1$ .

$4 n^{6} + 16 n^{3} m^{2} + 1 \cdot {(4 m^{2})}^{2}$

Step 4.20

Multiply ${(4 m^{2})}^{2}$ by $1$ .

$4 n^{6} + 16 n^{3} m^{2} + {(4 m^{2})}^{2}$

Step 4.21

Apply the product rule to $4 m^{2}$ .

$4 n^{6} + 16 n^{3} m^{2} + 4^{2} {(m^{2})}^{2}$

Step 4.22

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

$4 n^{6} + 16 n^{3} m^{2} + 16 {(m^{2})}^{2}$

Step 4.23

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

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Step 4.23.1

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

$4 n^{6} + 16 n^{3} m^{2} + 16 m^{2 \cdot 2}$

Step 4.23.2

Multiply $2$ by $2$ .

$4 n^{6} + 16 n^{3} m^{2} + 16 m^{4}$