Algebra Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify each term.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

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

$27 {(x^{3})}^{3} y^{3} + 3 {(3 x^{3} y)}^{2} (- 2 z^{4}) + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.3

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

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Step 2.3.1

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

$27 x^{3 \cdot 3} y^{3} + 3 {(3 x^{3} y)}^{2} (- 2 z^{4}) + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.3.2

Multiply $3$ by $3$ .

$27 x^{9} y^{3} + 3 {(3 x^{3} y)}^{2} (- 2 z^{4}) + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.4

Rewrite using the commutative property of multiplication.

$27 x^{9} y^{3} + 3 \cdot - 2 {(3 x^{3} y)}^{2} z^{4} + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.5

Multiply $3$ by $- 2$ .

$27 x^{9} y^{3} - 6 {(3 x^{3} y)}^{2} z^{4} + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.6

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

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Step 2.6.1

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

$27 x^{9} y^{3} - 6 ({(3 x^{3})}^{2} y^{2}) z^{4} + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.6.2

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

$27 x^{9} y^{3} - 6 (3^{2} {(x^{3})}^{2} y^{2}) z^{4} + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.7

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

$27 x^{9} y^{3} - 6 (9 {(x^{3})}^{2} y^{2}) z^{4} + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.8

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

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Step 2.8.1

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

$27 x^{9} y^{3} - 6 (9 x^{3 \cdot 2} y^{2}) z^{4} + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.8.2

Multiply $3$ by $2$ .

$27 x^{9} y^{3} - 6 (9 x^{6} y^{2}) z^{4} + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.9

Multiply $9$ by $- 6$ .

$27 x^{9} y^{3} - 54 (x^{6} y^{2}) z^{4} + 3 (3 x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.10

Multiply $3$ by $3$ .

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 9 (x^{3} y) {(- 2 z^{4})}^{2} + {(- 2 z^{4})}^{3}$

Step 2.11

Apply the product rule to $- 2 z^{4}$ .

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 9 x^{3} y ({(- 2)}^{2} {(z^{4})}^{2}) + {(- 2 z^{4})}^{3}$

Step 2.12

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

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 9 x^{3} y (4 {(z^{4})}^{2}) + {(- 2 z^{4})}^{3}$

Step 2.13

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

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Step 2.13.1

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

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 9 x^{3} y (4 z^{4 \cdot 2}) + {(- 2 z^{4})}^{3}$

Step 2.13.2

Multiply $4$ by $2$ .

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 9 x^{3} y (4 z^{8}) + {(- 2 z^{4})}^{3}$

Step 2.14

Multiply $4$ by $9$ .

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 36 x^{3} y z^{8} + {(- 2 z^{4})}^{3}$

Step 2.15

Apply the product rule to $- 2 z^{4}$ .

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 36 x^{3} y z^{8} + {(- 2)}^{3} {(z^{4})}^{3}$

Step 2.16

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

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 36 x^{3} y z^{8} - 8 {(z^{4})}^{3}$

Step 2.17

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

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Step 2.17.1

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

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 36 x^{3} y z^{8} - 8 z^{4 \cdot 3}$

Step 2.17.2

Multiply $4$ by $3$ .

$27 x^{9} y^{3} - 54 x^{6} y^{2} z^{4} + 36 x^{3} y z^{8} - 8 z^{12}$