Algebra Examples

${(4 x^{3} + 7 y^{3} z^{4})}^{2}$

Step 1

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

$\sum_{k = 0}^{2} \frac{2!}{(2 - k)! k!} \cdot {(4 x^{3})}^{2 - k} \cdot {(7 y^{3} z^{4})}^{k}$

Step 2

Expand the summation.

$\frac{2!}{(2 - 0)! 0!} \cdot {(4 x^{3})}^{2 - 0} \cdot {(7 y^{3} z^{4})}^{0} + \frac{2!}{(2 - 1)! 1!} \cdot {(4 x^{3})}^{2 - 1} \cdot {(7 y^{3} z^{4})}^{1} + \frac{2!}{(2 - 2)! 2!} \cdot {(4 x^{3})}^{2 - 2} \cdot {(7 y^{3} z^{4})}^{2}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(4 x^{3})}^{2} \cdot {(7 y^{3} z^{4})}^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4

Simplify each term.

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Step 4.1

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

${(4 x^{3})}^{2} \cdot {(7 y^{3} z^{4})}^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.2

Apply the product rule to $4 x^{3}$ .

$4^{2} {(x^{3})}^{2} \cdot {(7 y^{3} z^{4})}^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.3

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

$16 {(x^{3})}^{2} \cdot {(7 y^{3} z^{4})}^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.4

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

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Step 4.4.1

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

$16 x^{3 \cdot 2} \cdot {(7 y^{3} z^{4})}^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.4.2

Multiply $3$ by $2$ .

$16 x^{6} \cdot {(7 y^{3} z^{4})}^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.5

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.5.1

Apply the product rule to $7 y^{3} z^{4}$ .

$16 x^{6} \cdot ({(7 y^{3})}^{0} {(z^{4})}^{0}) + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.5.2

Apply the product rule to $7 y^{3}$ .

$16 x^{6} \cdot (7^{0} {(y^{3})}^{0} {(z^{4})}^{0}) + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.6

Anything raised to $0$ is $1$ .

$16 x^{6} \cdot (1 {(y^{3})}^{0} {(z^{4})}^{0}) + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.7

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

$16 x^{6} \cdot ({(y^{3})}^{0} {(z^{4})}^{0}) + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

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

$16 x^{6} \cdot (y^{3 \cdot 0} {(z^{4})}^{0}) + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.8.2

Multiply $3$ by $0$ .

$16 x^{6} \cdot (y^{0} {(z^{4})}^{0}) + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.9

Anything raised to $0$ is $1$ .

$16 x^{6} \cdot (1 {(z^{4})}^{0}) + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.10

Rewrite using the commutative property of multiplication.

$16 \cdot 1 x^{6} {(z^{4})}^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.11

Multiply $16$ by $1$ .

$16 x^{6} {(z^{4})}^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.12

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

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Step 4.12.1

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

$16 x^{6} z^{4 \cdot 0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.12.2

Multiply $4$ by $0$ .

$16 x^{6} z^{0} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.13

Anything raised to $0$ is $1$ .

$16 x^{6} \cdot 1 + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.14

Multiply $16$ by $1$ .

$16 x^{6} + 2 \cdot {(4 x^{3})}^{1} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.15

Simplify.

$16 x^{6} + 2 \cdot (4 x^{3}) \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.16

Multiply $4$ by $2$ .

$16 x^{6} + 8 x^{3} \cdot {(7 y^{3} z^{4})}^{1} + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.17

Simplify.

$16 x^{6} + 8 x^{3} \cdot (7 y^{3} z^{4}) + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.18

Multiply $7$ by $8$ .

$16 x^{6} + 56 x^{3} (y^{3} z^{4}) + 1 \cdot {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.19

Multiply ${(4 x^{3})}^{0}$ by $1$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + {(4 x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.20

Apply the product rule to $4 x^{3}$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 4^{0} {(x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.21

Anything raised to $0$ is $1$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 1 {(x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.22

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

$16 x^{6} + 56 x^{3} y^{3} z^{4} + {(x^{3})}^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.23

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

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Step 4.23.1

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

$16 x^{6} + 56 x^{3} y^{3} z^{4} + x^{3 \cdot 0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.23.2

Multiply $3$ by $0$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + x^{0} \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.24

Anything raised to $0$ is $1$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 1 \cdot {(7 y^{3} z^{4})}^{2}$

Step 4.25

Multiply ${(7 y^{3} z^{4})}^{2}$ by $1$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + {(7 y^{3} z^{4})}^{2}$

Step 4.26

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.26.1

Apply the product rule to $7 y^{3} z^{4}$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + {(7 y^{3})}^{2} {(z^{4})}^{2}$

Step 4.26.2

Apply the product rule to $7 y^{3}$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 7^{2} {(y^{3})}^{2} {(z^{4})}^{2}$

Step 4.27

Raise $7$ to the power of $2$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 49 {(y^{3})}^{2} {(z^{4})}^{2}$

Step 4.28

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

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Step 4.28.1

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

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 49 y^{3 \cdot 2} {(z^{4})}^{2}$

Step 4.28.2

Multiply $3$ by $2$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 49 y^{6} {(z^{4})}^{2}$

Step 4.29

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

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Step 4.29.1

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

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 49 y^{6} z^{4 \cdot 2}$

Step 4.29.2

Multiply $4$ by $2$ .

$16 x^{6} + 56 x^{3} y^{3} z^{4} + 49 y^{6} z^{8}$