Precalculus Examples

${(x^{2} y - 1)}^{4}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

The triangle can be used to calculate the coefficients of the expansion of ${(a + b)}^{n}$ by taking the exponent $n$ and adding $1$ . The coefficients will correspond with line $n + 1$ of the triangle. For ${(x^{2} y - 1)}^{4}$ , $n = 4$ so the coefficients of the expansion will correspond with line $5$ .

Step 2

The expansion follows the rule ${(a + b)}^{n} = c_{0} a^{n} b^{0} + c_{1} a^{n - 1} b^{1} + c_{n - 1} a^{1} b^{n - 1} + c_{n} a^{0} b^{n}$ . The values of the coefficients, from the triangle, are $1 - 4 - 6 - 4 - 1$ .

$1 a^{4} b^{0} + 4 a^{3} b + 6 a^{2} b^{2} + 4 a b^{3} + 1 a^{0} b^{4}$

Step 3

Substitute the actual values of $a$ $x^{2} y$ and $b$ $- 1$ into the expression.

$1 {(x^{2} y)}^{4} {(- 1)}^{0} + 4 {(x^{2} y)}^{3} {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4

Simplify each term.

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Step 4.1

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

${(x^{2} y)}^{4} {(- 1)}^{0} + 4 {(x^{2} y)}^{3} {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.2

Apply the product rule to $x^{2} y$ .

${(x^{2})}^{4} y^{4} {(- 1)}^{0} + 4 {(x^{2} y)}^{3} {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.3

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

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Step 4.3.1

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

$x^{2 \cdot 4} y^{4} {(- 1)}^{0} + 4 {(x^{2} y)}^{3} {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.3.2

Multiply $2$ by $4$ .

$x^{8} y^{4} {(- 1)}^{0} + 4 {(x^{2} y)}^{3} {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.4

Anything raised to $0$ is $1$ .

$x^{8} y^{4} \cdot 1 + 4 {(x^{2} y)}^{3} {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.5

Multiply $x^{8}$ by $1$ .

$x^{8} y^{4} + 4 {(x^{2} y)}^{3} {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.6

Apply the product rule to $x^{2} y$ .

$x^{8} y^{4} + 4 ({(x^{2})}^{3} y^{3}) {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.7

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

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Step 4.7.1

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

$x^{8} y^{4} + 4 (x^{2 \cdot 3} y^{3}) {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.7.2

Multiply $2$ by $3$ .

$x^{8} y^{4} + 4 (x^{6} y^{3}) {(- 1)}^{1} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.8

Evaluate the exponent.

$x^{8} y^{4} + 4 x^{6} y^{3} \cdot - 1 + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.9

Multiply $- 1$ by $4$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 {(x^{2} y)}^{2} {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.10

Apply the product rule to $x^{2} y$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 ({(x^{2})}^{2} y^{2}) {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.11

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

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Step 4.11.1

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

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 (x^{2 \cdot 2} y^{2}) {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.11.2

Multiply $2$ by $2$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 (x^{4} y^{2}) {(- 1)}^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.12

Raise $- 1$ to the power of $2$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} \cdot 1 + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.13

Multiply $6$ by $1$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} + 4 {(x^{2} y)}^{1} {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.14

Simplify.

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} + 4 (x^{2} y) {(- 1)}^{3} + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.15

Raise $- 1$ to the power of $3$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} + 4 x^{2} y \cdot - 1 + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.16

Multiply $- 1$ by $4$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + 1 {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.17

Multiply ${(x^{2} y)}^{0}$ by $1$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + {(x^{2} y)}^{0} {(- 1)}^{4}$

Step 4.18

Apply the product rule to $x^{2} y$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + {(x^{2})}^{0} y^{0} {(- 1)}^{4}$

Step 4.19

Multiply the exponents in ${(x^{2})}^{0}$ .

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Step 4.19.1

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

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + x^{2 \cdot 0} y^{0} {(- 1)}^{4}$

Step 4.19.2

Multiply $2$ by $0$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + x^{0} y^{0} {(- 1)}^{4}$

Step 4.20

Anything raised to $0$ is $1$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + 1 y^{0} {(- 1)}^{4}$

Step 4.21

Multiply $y^{0}$ by $1$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + y^{0} {(- 1)}^{4}$

Step 4.22

Anything raised to $0$ is $1$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + 1 {(- 1)}^{4}$

Step 4.23

Multiply ${(- 1)}^{4}$ by $1$ .

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + {(- 1)}^{4}$

Step 4.24

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

$x^{8} y^{4} - 4 x^{6} y^{3} + 6 x^{4} y^{2} - 4 x^{2} y + 1$