Calculus Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

Multiply $2$ by $4$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $3$ .

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

Step 2.3

Multiply $- 4$ by $4$ .

$x^{8} - 16 x^{6} + 6 {(x^{2})}^{2} {(- 4)}^{2} + 4 x^{2} {(- 4)}^{3} + {(- 4)}^{4}$

Step 2.4

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

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Step 2.4.1

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

$x^{8} - 16 x^{6} + 6 x^{2 \cdot 2} {(- 4)}^{2} + 4 x^{2} {(- 4)}^{3} + {(- 4)}^{4}$

Step 2.4.2

Multiply $2$ by $2$ .

$x^{8} - 16 x^{6} + 6 x^{4} {(- 4)}^{2} + 4 x^{2} {(- 4)}^{3} + {(- 4)}^{4}$

Step 2.5

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

$x^{8} - 16 x^{6} + 6 x^{4} \cdot 16 + 4 x^{2} {(- 4)}^{3} + {(- 4)}^{4}$

Step 2.6

Multiply $16$ by $6$ .

$x^{8} - 16 x^{6} + 96 x^{4} + 4 x^{2} {(- 4)}^{3} + {(- 4)}^{4}$

Step 2.7

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

$x^{8} - 16 x^{6} + 96 x^{4} + 4 x^{2} \cdot - 64 + {(- 4)}^{4}$

Step 2.8

Multiply $- 64$ by $4$ .

$x^{8} - 16 x^{6} + 96 x^{4} - 256 x^{2} + {(- 4)}^{4}$

Step 2.9

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

$x^{8} - 16 x^{6} + 96 x^{4} - 256 x^{2} + 256$