Precalculus Examples

${(x + 2 y)}^{6}$

Step 1

Pascal's Triangle can be displayed as such:

$1$

$1 - 1$

$1 - 2 - 1$

$1 - 3 - 3 - 1$

$1 - 4 - 6 - 4 - 1$

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

$1 - 6 - 15 - 20 - 15 - 6 - 1$

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 ${(x + 2 y)}^{6}$ , $n = 6$ so the coefficients of the expansion will correspond with line $7$ .

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 $1 - 6 - 15 - 20 - 15 - 6 - 1$ .

$1 a^{6} b^{0} + 6 a^{5} b + 15 a^{4} b^{2} + 20 a^{3} b^{3} + 15 a^{2} b^{4} + 6 a b^{5} + 1 a^{0} b^{6}$

Step 3

Substitute the actual values of $a$ $x$ and $b$ $2 y$ into the expression.

$1 {(x)}^{6} {(2 y)}^{0} + 6 {(x)}^{5} {(2 y)}^{1} + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4

Simplify each term.

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Step 4.1

Multiply ${(x)}^{6}$ by $1$ .

${(x)}^{6} {(2 y)}^{0} + 6 {(x)}^{5} {(2 y)}^{1} + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.2

Apply the product rule to $2 y$ .

$x^{6} (2^{0} y^{0}) + 6 {(x)}^{5} {(2 y)}^{1} + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.3

Rewrite using the commutative property of multiplication.

$2^{0} x^{6} y^{0} + 6 {(x)}^{5} {(2 y)}^{1} + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.4

Anything raised to $0$ is $1$ .

$1 x^{6} y^{0} + 6 {(x)}^{5} {(2 y)}^{1} + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.5

Multiply $x^{6}$ by $1$ .

$x^{6} y^{0} + 6 {(x)}^{5} {(2 y)}^{1} + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.6

Anything raised to $0$ is $1$ .

$x^{6} \cdot 1 + 6 {(x)}^{5} {(2 y)}^{1} + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.7

Multiply $x^{6}$ by $1$ .

$x^{6} + 6 {(x)}^{5} {(2 y)}^{1} + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.8

Simplify.

$x^{6} + 6 x^{5} (2 y) + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.9

Rewrite using the commutative property of multiplication.

$x^{6} + 6 \cdot 2 x^{5} y + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.10

Multiply $6$ by $2$ .

$x^{6} + 12 x^{5} y + 15 {(x)}^{4} {(2 y)}^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.11

Apply the product rule to $2 y$ .

$x^{6} + 12 x^{5} y + 15 x^{4} (2^{2} y^{2}) + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.12

Rewrite using the commutative property of multiplication.

$x^{6} + 12 x^{5} y + 15 \cdot 2^{2} x^{4} y^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.13

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

$x^{6} + 12 x^{5} y + 15 \cdot 4 x^{4} y^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.14

Multiply $15$ by $4$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 20 {(x)}^{3} {(2 y)}^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.15

Apply the product rule to $2 y$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 20 x^{3} (2^{3} y^{3}) + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.16

Rewrite using the commutative property of multiplication.

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 20 \cdot 2^{3} x^{3} y^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.17

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

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 20 \cdot 8 x^{3} y^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.18

Multiply $20$ by $8$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 15 {(x)}^{2} {(2 y)}^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.19

Apply the product rule to $2 y$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 15 x^{2} (2^{4} y^{4}) + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.20

Rewrite using the commutative property of multiplication.

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 15 \cdot 2^{4} x^{2} y^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.21

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

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 15 \cdot 16 x^{2} y^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.22

Multiply $15$ by $16$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 6 {(x)}^{1} {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.23

Simplify.

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 6 x {(2 y)}^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.24

Apply the product rule to $2 y$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 6 x (2^{5} y^{5}) + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.25

Rewrite using the commutative property of multiplication.

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 6 \cdot 2^{5} x y^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.26

Raise $2$ to the power of $5$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 6 \cdot 32 x y^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.27

Multiply $6$ by $32$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 192 x y^{5} + 1 {(x)}^{0} {(2 y)}^{6}$

Step 4.28

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

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 192 x y^{5} + {(x)}^{0} {(2 y)}^{6}$

Step 4.29

Anything raised to $0$ is $1$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 192 x y^{5} + 1 {(2 y)}^{6}$

Step 4.30

Multiply ${(2 y)}^{6}$ by $1$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 192 x y^{5} + {(2 y)}^{6}$

Step 4.31

Apply the product rule to $2 y$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 192 x y^{5} + 2^{6} y^{6}$

Step 4.32

Raise $2$ to the power of $6$ .

$x^{6} + 12 x^{5} y + 60 x^{4} y^{2} + 160 x^{3} y^{3} + 240 x^{2} y^{4} + 192 x y^{5} + 64 y^{6}$