Algebra Examples

${(- 3 x^{2} + y)}^{5}$

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 ${(- 3 x^{2} + y)}^{5}$ , $n = 5$ 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 $a$ $- 3 x^{2}$ and $b$ $y$ into the expression.

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

Step 4

Simplify each term.

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Step 4.1

Multiply ${(- 3 x^{2})}^{5}$ by $1$ .

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

Step 4.2

Apply the product rule to $- 3 x^{2}$ .

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

Step 4.3

Raise $- 3$ to the power of $5$ .

$- 243 {(x^{2})}^{5} {(y)}^{0} + 5 {(- 3 x^{2})}^{4} {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.4

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

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Step 4.4.1

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

$- 243 x^{2 \cdot 5} {(y)}^{0} + 5 {(- 3 x^{2})}^{4} {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.4.2

Multiply $2$ by $5$ .

$- 243 x^{10} {(y)}^{0} + 5 {(- 3 x^{2})}^{4} {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.5

Anything raised to $0$ is $1$ .

$- 243 x^{10} \cdot 1 + 5 {(- 3 x^{2})}^{4} {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.6

Multiply $- 243$ by $1$ .

$- 243 x^{10} + 5 {(- 3 x^{2})}^{4} {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.7

Apply the product rule to $- 3 x^{2}$ .

$- 243 x^{10} + 5 ({(- 3)}^{4} {(x^{2})}^{4}) {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.8

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

$- 243 x^{10} + 5 (81 {(x^{2})}^{4}) {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.9

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

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Step 4.9.1

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

$- 243 x^{10} + 5 (81 x^{2 \cdot 4}) {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.9.2

Multiply $2$ by $4$ .

$- 243 x^{10} + 5 (81 x^{8}) {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.10

Multiply $81$ by $5$ .

$- 243 x^{10} + 405 x^{8} {(y)}^{1} + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.11

Simplify.

$- 243 x^{10} + 405 x^{8} y + 10 {(- 3 x^{2})}^{3} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.12

Apply the product rule to $- 3 x^{2}$ .

$- 243 x^{10} + 405 x^{8} y + 10 ({(- 3)}^{3} {(x^{2})}^{3}) {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.13

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

$- 243 x^{10} + 405 x^{8} y + 10 (- 27 {(x^{2})}^{3}) {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.14

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

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Step 4.14.1

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

$- 243 x^{10} + 405 x^{8} y + 10 (- 27 x^{2 \cdot 3}) {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.14.2

Multiply $2$ by $3$ .

$- 243 x^{10} + 405 x^{8} y + 10 (- 27 x^{6}) {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.15

Multiply $- 27$ by $10$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} {(y)}^{2} + 10 {(- 3 x^{2})}^{2} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.16

Apply the product rule to $- 3 x^{2}$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 10 ({(- 3)}^{2} {(x^{2})}^{2}) {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.17

Raise $- 3$ to the power of $2$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 10 (9 {(x^{2})}^{2}) {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.18

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

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Step 4.18.1

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

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 10 (9 x^{2 \cdot 2}) {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.18.2

Multiply $2$ by $2$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 10 (9 x^{4}) {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.19

Multiply $9$ by $10$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} {(y)}^{3} + 5 {(- 3 x^{2})}^{1} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.20

Simplify.

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} + 5 (- 3 x^{2}) {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.21

Multiply $- 3$ by $5$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} {(y)}^{4} + 1 {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.22

Multiply ${(- 3 x^{2})}^{0}$ by $1$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} y^{4} + {(- 3 x^{2})}^{0} {(y)}^{5}$

Step 4.23

Apply the product rule to $- 3 x^{2}$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} y^{4} + {(- 3)}^{0} {(x^{2})}^{0} {(y)}^{5}$

Step 4.24

Anything raised to $0$ is $1$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} y^{4} + 1 {(x^{2})}^{0} {(y)}^{5}$

Step 4.25

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

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} y^{4} + {(x^{2})}^{0} {(y)}^{5}$

Step 4.26

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

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Step 4.26.1

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

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} y^{4} + x^{2 \cdot 0} {(y)}^{5}$

Step 4.26.2

Multiply $2$ by $0$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} y^{4} + x^{0} {(y)}^{5}$

Step 4.27

Anything raised to $0$ is $1$ .

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} y^{4} + 1 {(y)}^{5}$

Step 4.28

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

$- 243 x^{10} + 405 x^{8} y - 270 x^{6} y^{2} + 90 x^{4} y^{3} - 15 x^{2} y^{4} + y^{5}$