Expand the Trigonometric Expression (-4a^2+b)^5

${(- 4 a^{2} + b)}^{5}$

Step 1

Use the Binomial Theorem.

${(- 4 a^{2})}^{5} + 5 {(- 4 a^{2})}^{4} b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Step 2

Simplify each term.

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

${(- 4)}^{5} {(a^{2})}^{5} + 5 {(- 4 a^{2})}^{4} b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Raise $- 4$ to the power of $5$ .

$- 1024 {(a^{2})}^{5} + 5 {(- 4 a^{2})}^{4} b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply the exponents in ${(a^{2})}^{5}$ .

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

$- 1024 a^{2 \cdot 5} + 5 {(- 4 a^{2})}^{4} b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply $2$ by $5$ .

$- 1024 a^{10} + 5 {(- 4 a^{2})}^{4} b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

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

$- 1024 a^{10} + 5 ({(- 4)}^{4} {(a^{2})}^{4}) b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

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

$- 1024 a^{10} + 5 (256 {(a^{2})}^{4}) b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

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

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

$- 1024 a^{10} + 5 (256 a^{2 \cdot 4}) b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply $2$ by $4$ .

$- 1024 a^{10} + 5 (256 a^{8}) b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply $256$ by $5$ .

$- 1024 a^{10} + 1280 a^{8} b + 10 {(- 4 a^{2})}^{3} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

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

$- 1024 a^{10} + 1280 a^{8} b + 10 ({(- 4)}^{3} {(a^{2})}^{3}) b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Raise $- 4$ to the power of $3$ .

$- 1024 a^{10} + 1280 a^{8} b + 10 (- 64 {(a^{2})}^{3}) b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

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

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

$- 1024 a^{10} + 1280 a^{8} b + 10 (- 64 a^{2 \cdot 3}) b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply $2$ by $3$ .

$- 1024 a^{10} + 1280 a^{8} b + 10 (- 64 a^{6}) b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply $- 64$ by $10$ .

$- 1024 a^{10} + 1280 a^{8} b - 640 a^{6} b^{2} + 10 {(- 4 a^{2})}^{2} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

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

$- 1024 a^{10} + 1280 a^{8} b - 640 a^{6} b^{2} + 10 ({(- 4)}^{2} {(a^{2})}^{2}) b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

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

$- 1024 a^{10} + 1280 a^{8} b - 640 a^{6} b^{2} + 10 (16 {(a^{2})}^{2}) b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

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

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

$- 1024 a^{10} + 1280 a^{8} b - 640 a^{6} b^{2} + 10 (16 a^{2 \cdot 2}) b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply $2$ by $2$ .

$- 1024 a^{10} + 1280 a^{8} b - 640 a^{6} b^{2} + 10 (16 a^{4}) b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply $16$ by $10$ .

$- 1024 a^{10} + 1280 a^{8} b - 640 a^{6} b^{2} + 160 a^{4} b^{3} + 5 (- 4 a^{2}) b^{4} + b^{5}$

Multiply $- 4$ by $5$ .

$- 1024 a^{10} + 1280 a^{8} b - 640 a^{6} b^{2} + 160 a^{4} b^{3} - 20 a^{2} b^{4} + b^{5}$

Expand the Trigonometric Expression (-4a^2+b)^5

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