Expand the Trigonometric Expression (-4u^2+v)^5

${(- 4 u^{2} + v)}^{5}$

Step 1

Use the Binomial Theorem.

${(- 4 u^{2})}^{5} + 5 {(- 4 u^{2})}^{4} v + 10 {(- 4 u^{2})}^{3} v^{2} + 10 {(- 4 u^{2})}^{2} v^{3} + 5 (- 4 u^{2}) v^{4} + v^{5}$

Step 2

Simplify each term.

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

${(- 4)}^{5} {(u^{2})}^{5} + 5 {(- 4 u^{2})}^{4} v + 10 {(- 4 u^{2})}^{3} v^{2} + 10 {(- 4 u^{2})}^{2} v^{3} + 5 (- 4 u^{2}) v^{4} + v^{5}$

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

$- 1024 {(u^{2})}^{5} + 5 {(- 4 u^{2})}^{4} v + 10 {(- 4 u^{2})}^{3} v^{2} + 10 {(- 4 u^{2})}^{2} v^{3} + 5 (- 4 u^{2}) v^{4} + v^{5}$

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

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

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

Multiply $2$ by $5$ .

$- 1024 u^{10} + 5 {(- 4 u^{2})}^{4} v + 10 {(- 4 u^{2})}^{3} v^{2} + 10 {(- 4 u^{2})}^{2} v^{3} + 5 (- 4 u^{2}) v^{4} + v^{5}$

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

$- 1024 u^{10} + 5 ({(- 4)}^{4} {(u^{2})}^{4}) v + 10 {(- 4 u^{2})}^{3} v^{2} + 10 {(- 4 u^{2})}^{2} v^{3} + 5 (- 4 u^{2}) v^{4} + v^{5}$

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

$- 1024 u^{10} + 5 (256 {(u^{2})}^{4}) v + 10 {(- 4 u^{2})}^{3} v^{2} + 10 {(- 4 u^{2})}^{2} v^{3} + 5 (- 4 u^{2}) v^{4} + v^{5}$

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

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

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

Multiply $2$ by $4$ .

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

Multiply $256$ by $5$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $3$ .

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

Multiply $- 64$ by $10$ .

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

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

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

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

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

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

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

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

Multiply $2$ by $2$ .

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

Multiply $16$ by $10$ .

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

Multiply $- 4$ by $5$ .

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

Expand the Trigonometric Expression (-4u^2+v)^5

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