Find the Third Term (x^2+2)^4

${(x^{2} + 2)}^{4}$

Step 1

Substitute in the value of $n$ to find the $n$ th term.

$a_{3} = {(x^{2} + 2)}^{4}$

Step 2

Use the Binomial Theorem.

$a_{3} = {(x^{2})}^{4} + 4 {(x^{2})}^{3} \cdot 2 + 6 {(x^{2})}^{2} \cdot 2^{2} + 4 x^{2} \cdot 2^{3} + 2^{4}$

Step 3

Simplify each term.

Multiply the exponents in ${(x^{2})}^{4}$ .

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

$a_{3} = x^{2 \cdot 4} + 4 {(x^{2})}^{3} \cdot 2 + 6 {(x^{2})}^{2} \cdot 2^{2} + 4 x^{2} \cdot 2^{3} + 2^{4}$

Multiply $2$ by $4$ .

$a_{3} = x^{8} + 4 {(x^{2})}^{3} \cdot 2 + 6 {(x^{2})}^{2} \cdot 2^{2} + 4 x^{2} \cdot 2^{3} + 2^{4}$

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

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

$a_{3} = x^{8} + 4 x^{2 \cdot 3} \cdot 2 + 6 {(x^{2})}^{2} \cdot 2^{2} + 4 x^{2} \cdot 2^{3} + 2^{4}$

Multiply $2$ by $3$ .

$a_{3} = x^{8} + 4 x^{6} \cdot 2 + 6 {(x^{2})}^{2} \cdot 2^{2} + 4 x^{2} \cdot 2^{3} + 2^{4}$

Multiply $2$ by $4$ .

$a_{3} = x^{8} + 8 x^{6} + 6 {(x^{2})}^{2} \cdot 2^{2} + 4 x^{2} \cdot 2^{3} + 2^{4}$

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

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

$a_{3} = x^{8} + 8 x^{6} + 6 x^{2 \cdot 2} \cdot 2^{2} + 4 x^{2} \cdot 2^{3} + 2^{4}$

Multiply $2$ by $2$ .

$a_{3} = x^{8} + 8 x^{6} + 6 x^{4} \cdot 2^{2} + 4 x^{2} \cdot 2^{3} + 2^{4}$

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

$a_{3} = x^{8} + 8 x^{6} + 6 x^{4} \cdot 4 + 4 x^{2} \cdot 2^{3} + 2^{4}$

Multiply $4$ by $6$ .

$a_{3} = x^{8} + 8 x^{6} + 24 x^{4} + 4 x^{2} \cdot 2^{3} + 2^{4}$

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

$a_{3} = x^{8} + 8 x^{6} + 24 x^{4} + 4 x^{2} \cdot 8 + 2^{4}$

Multiply $8$ by $4$ .

$a_{3} = x^{8} + 8 x^{6} + 24 x^{4} + 32 x^{2} + 2^{4}$

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

$a_{3} = x^{8} + 8 x^{6} + 24 x^{4} + 32 x^{2} + 16$