Precalculus Examples

${(x^{2} y - 1)}^{5}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

$1 - 5 - 10 - 10 - 5 - 1$

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 ${(x^{2} y - 1)}^{5}$ , $n = 5$ so the coefficients of the expansion will correspond with line $6$ .

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 $1 - 5 - 10 - 10 - 5 - 1$ .

$1 a^{5} b^{0} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 a^{0} b^{5}$

Step 3

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

$1 {(x^{2} y)}^{5} {(- 1)}^{0} + 5 {(x^{2} y)}^{4} {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(x^{2} y)}^{5}$ by $1$ .

${(x^{2} y)}^{5} {(- 1)}^{0} + 5 {(x^{2} y)}^{4} {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.2

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

${(x^{2})}^{5} y^{5} {(- 1)}^{0} + 5 {(x^{2} y)}^{4} {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.3

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

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Step 4.3.1

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

$x^{2 \cdot 5} y^{5} {(- 1)}^{0} + 5 {(x^{2} y)}^{4} {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.3.2

Multiply $2$ by $5$ .

$x^{10} y^{5} {(- 1)}^{0} + 5 {(x^{2} y)}^{4} {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.4

Anything raised to $0$ is $1$ .

$x^{10} y^{5} \cdot 1 + 5 {(x^{2} y)}^{4} {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.5

Multiply $x^{10}$ by $1$ .

$x^{10} y^{5} + 5 {(x^{2} y)}^{4} {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.6

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

$x^{10} y^{5} + 5 ({(x^{2})}^{4} y^{4}) {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.7

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

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Step 4.7.1

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

$x^{10} y^{5} + 5 (x^{2 \cdot 4} y^{4}) {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.7.2

Multiply $2$ by $4$ .

$x^{10} y^{5} + 5 (x^{8} y^{4}) {(- 1)}^{1} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.8

Evaluate the exponent.

$x^{10} y^{5} + 5 x^{8} y^{4} \cdot - 1 + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.9

Multiply $- 1$ by $5$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 {(x^{2} y)}^{3} {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.10

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 ({(x^{2})}^{3} y^{3}) {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.11

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

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Step 4.11.1

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 (x^{2 \cdot 3} y^{3}) {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.11.2

Multiply $2$ by $3$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 (x^{6} y^{3}) {(- 1)}^{2} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.12

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} \cdot 1 + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.13

Multiply $10$ by $1$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} + 10 {(x^{2} y)}^{2} {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.14

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} + 10 ({(x^{2})}^{2} y^{2}) {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.15

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

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Step 4.15.1

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} + 10 (x^{2 \cdot 2} y^{2}) {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.15.2

Multiply $2$ by $2$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} + 10 (x^{4} y^{2}) {(- 1)}^{3} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.16

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} + 10 x^{4} y^{2} \cdot - 1 + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.17

Multiply $- 1$ by $10$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 {(x^{2} y)}^{1} {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.18

Simplify.

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 (x^{2} y) {(- 1)}^{4} + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.19

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y \cdot 1 + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.20

Multiply $5$ by $1$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + 1 {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.21

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + {(x^{2} y)}^{0} {(- 1)}^{5}$

Step 4.22

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + {(x^{2})}^{0} y^{0} {(- 1)}^{5}$

Step 4.23

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

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Step 4.23.1

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + x^{2 \cdot 0} y^{0} {(- 1)}^{5}$

Step 4.23.2

Multiply $2$ by $0$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + x^{0} y^{0} {(- 1)}^{5}$

Step 4.24

Anything raised to $0$ is $1$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + 1 y^{0} {(- 1)}^{5}$

Step 4.25

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

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + y^{0} {(- 1)}^{5}$

Step 4.26

Anything raised to $0$ is $1$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + 1 {(- 1)}^{5}$

Step 4.27

Multiply ${(- 1)}^{5}$ by $1$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y + {(- 1)}^{5}$

Step 4.28

Raise $- 1$ to the power of $5$ .

$x^{10} y^{5} - 5 x^{8} y^{4} + 10 x^{6} y^{3} - 10 x^{4} y^{2} + 5 x^{2} y - 1$