Algebra Examples

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

Step 4

Simplify each term.

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Step 4.1

Multiply ${(3 x^{2})}^{4}$ by $1$ .

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

Step 4.2

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

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

Step 4.3

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

$81 {(x^{2})}^{4} \cdot {(- 4 y^{4})}^{0} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.4

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

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Step 4.4.1

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

$81 x^{2 \cdot 4} \cdot {(- 4 y^{4})}^{0} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.4.2

Multiply $2$ by $4$ .

$81 x^{8} \cdot {(- 4 y^{4})}^{0} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.5

Apply the product rule to $- 4 y^{4}$ .

$81 x^{8} \cdot ({(- 4)}^{0} {(y^{4})}^{0}) + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.6

Rewrite using the commutative property of multiplication.

$81 \cdot {(- 4)}^{0} x^{8} {(y^{4})}^{0} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.7

Anything raised to $0$ is $1$ .

$81 \cdot 1 x^{8} {(y^{4})}^{0} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.8

Multiply $81$ by $1$ .

$81 x^{8} {(y^{4})}^{0} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.9

Multiply the exponents in ${(y^{4})}^{0}$ .

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Step 4.9.1

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

$81 x^{8} y^{4 \cdot 0} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.9.2

Multiply $4$ by $0$ .

$81 x^{8} y^{0} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.10

Anything raised to $0$ is $1$ .

$81 x^{8} \cdot 1 + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.11

Multiply $81$ by $1$ .

$81 x^{8} + 4 \cdot {(3 x^{2})}^{3} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.12

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

$81 x^{8} + 4 \cdot (3^{3} {(x^{2})}^{3}) \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.13

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

$81 x^{8} + 4 \cdot (27 {(x^{2})}^{3}) \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

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, ${(a^{m})}^{n} = a^{m n}$ .

$81 x^{8} + 4 \cdot (27 x^{2 \cdot 3}) \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.14.2

Multiply $2$ by $3$ .

$81 x^{8} + 4 \cdot (27 x^{6}) \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.15

Multiply $27$ by $4$ .

$81 x^{8} + 108 x^{6} \cdot {(- 4 y^{4})}^{1} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.16

Simplify.

$81 x^{8} + 108 x^{6} \cdot (- 4 y^{4}) + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.17

Rewrite using the commutative property of multiplication.

$81 x^{8} + 108 \cdot - 4 x^{6} y^{4} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.18

Multiply $108$ by $- 4$ .

$81 x^{8} - 432 x^{6} y^{4} + 6 \cdot {(3 x^{2})}^{2} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.19

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

$81 x^{8} - 432 x^{6} y^{4} + 6 \cdot (3^{2} {(x^{2})}^{2}) \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.20

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

$81 x^{8} - 432 x^{6} y^{4} + 6 \cdot (9 {(x^{2})}^{2}) \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.21

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

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Step 4.21.1

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

$81 x^{8} - 432 x^{6} y^{4} + 6 \cdot (9 x^{2 \cdot 2}) \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.21.2

Multiply $2$ by $2$ .

$81 x^{8} - 432 x^{6} y^{4} + 6 \cdot (9 x^{4}) \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.22

Multiply $9$ by $6$ .

$81 x^{8} - 432 x^{6} y^{4} + 54 x^{4} \cdot {(- 4 y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.23

Apply the product rule to $- 4 y^{4}$ .

$81 x^{8} - 432 x^{6} y^{4} + 54 x^{4} \cdot ({(- 4)}^{2} {(y^{4})}^{2}) + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.24

Rewrite using the commutative property of multiplication.

$81 x^{8} - 432 x^{6} y^{4} + 54 \cdot {(- 4)}^{2} x^{4} {(y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.25

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

$81 x^{8} - 432 x^{6} y^{4} + 54 \cdot 16 x^{4} {(y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.26

Multiply $54$ by $16$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} {(y^{4})}^{2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.27

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

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Step 4.27.1

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{4 \cdot 2} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.27.2

Multiply $4$ by $2$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} + 4 \cdot {(3 x^{2})}^{1} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.28

Simplify.

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} + 4 \cdot (3 x^{2}) \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.29

Multiply $3$ by $4$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} + 12 x^{2} \cdot {(- 4 y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.30

Apply the product rule to $- 4 y^{4}$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} + 12 x^{2} \cdot ({(- 4)}^{3} {(y^{4})}^{3}) + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.31

Rewrite using the commutative property of multiplication.

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} + 12 \cdot {(- 4)}^{3} x^{2} {(y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.32

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} + 12 \cdot - 64 x^{2} {(y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.33

Multiply $12$ by $- 64$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} {(y^{4})}^{3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.34

Multiply the exponents in ${(y^{4})}^{3}$ .

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Step 4.34.1

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{4 \cdot 3} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.34.2

Multiply $4$ by $3$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + 1 \cdot {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.35

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + {(3 x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.36

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + 3^{0} {(x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.37

Anything raised to $0$ is $1$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + 1 {(x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.38

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + {(x^{2})}^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.39

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

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Step 4.39.1

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + x^{2 \cdot 0} \cdot {(- 4 y^{4})}^{4}$

Step 4.39.2

Multiply $2$ by $0$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + x^{0} \cdot {(- 4 y^{4})}^{4}$

Step 4.40

Anything raised to $0$ is $1$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + 1 \cdot {(- 4 y^{4})}^{4}$

Step 4.41

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + {(- 4 y^{4})}^{4}$

Step 4.42

Apply the product rule to $- 4 y^{4}$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + {(- 4)}^{4} {(y^{4})}^{4}$

Step 4.43

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + 256 {(y^{4})}^{4}$

Step 4.44

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

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Step 4.44.1

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

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + 256 y^{4 \cdot 4}$

Step 4.44.2

Multiply $4$ by $4$ .

$81 x^{8} - 432 x^{6} y^{4} + 864 x^{4} y^{8} - 768 x^{2} y^{12} + 256 y^{16}$