Algebra Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(4 x)}^{2 - k} \cdot {(- 6 y^{3})}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

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

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

Step 4.2

Apply the product rule to $4 x$ .

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

Step 4.3

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

$16 x^{2} \cdot {(- 6 y^{3})}^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.4

Apply the product rule to $- 6 y^{3}$ .

$16 x^{2} \cdot ({(- 6)}^{0} {(y^{3})}^{0}) + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.5

Rewrite using the commutative property of multiplication.

$16 \cdot {(- 6)}^{0} x^{2} {(y^{3})}^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.6

Anything raised to $0$ is $1$ .

$16 \cdot 1 x^{2} {(y^{3})}^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.7

Multiply $16$ by $1$ .

$16 x^{2} {(y^{3})}^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.8

Multiply the exponents in ${(y^{3})}^{0}$ .

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Step 4.8.1

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

$16 x^{2} y^{3 \cdot 0} + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.8.2

Multiply $3$ by $0$ .

$16 x^{2} y^{0} + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.9

Anything raised to $0$ is $1$ .

$16 x^{2} \cdot 1 + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.10

Multiply $16$ by $1$ .

$16 x^{2} + 2 \cdot {(4 x)}^{1} \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.11

Simplify.

$16 x^{2} + 2 \cdot (4 x) \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.12

Multiply $4$ by $2$ .

$16 x^{2} + 8 x \cdot {(- 6 y^{3})}^{1} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.13

Simplify.

$16 x^{2} + 8 x \cdot (- 6 y^{3}) + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.14

Rewrite using the commutative property of multiplication.

$16 x^{2} + 8 \cdot - 6 x y^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.15

Multiply $8$ by $- 6$ .

$16 x^{2} - 48 x y^{3} + 1 \cdot {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.16

Multiply ${(4 x)}^{0}$ by $1$ .

$16 x^{2} - 48 x y^{3} + {(4 x)}^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.17

Apply the product rule to $4 x$ .

$16 x^{2} - 48 x y^{3} + 4^{0} x^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.18

Anything raised to $0$ is $1$ .

$16 x^{2} - 48 x y^{3} + 1 x^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.19

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

$16 x^{2} - 48 x y^{3} + x^{0} \cdot {(- 6 y^{3})}^{2}$

Step 4.20

Anything raised to $0$ is $1$ .

$16 x^{2} - 48 x y^{3} + 1 \cdot {(- 6 y^{3})}^{2}$

Step 4.21

Multiply ${(- 6 y^{3})}^{2}$ by $1$ .

$16 x^{2} - 48 x y^{3} + {(- 6 y^{3})}^{2}$

Step 4.22

Apply the product rule to $- 6 y^{3}$ .

$16 x^{2} - 48 x y^{3} + {(- 6)}^{2} {(y^{3})}^{2}$

Step 4.23

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

$16 x^{2} - 48 x y^{3} + 36 {(y^{3})}^{2}$

Step 4.24

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

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Step 4.24.1

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

$16 x^{2} - 48 x y^{3} + 36 y^{3 \cdot 2}$

Step 4.24.2

Multiply $3$ by $2$ .

$16 x^{2} - 48 x y^{3} + 36 y^{6}$