Algebra Examples

${(5 x - y^{3})}^{3}$

Step 1

Use the Binomial Theorem.

${(5 x)}^{3} + 3 {(5 x)}^{2} (- y^{3}) + 3 (5 x) {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Apply the product rule to $5 x$ .

$5^{3} x^{3} + 3 {(5 x)}^{2} (- y^{3}) + 3 (5 x) {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.2

Raise $5$ to the power of $3$ .

$125 x^{3} + 3 {(5 x)}^{2} (- y^{3}) + 3 (5 x) {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.3

Rewrite using the commutative property of multiplication.

$125 x^{3} + 3 \cdot - 1 {(5 x)}^{2} y^{3} + 3 (5 x) {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.4

Multiply $3$ by $- 1$ .

$125 x^{3} - 3 {(5 x)}^{2} y^{3} + 3 (5 x) {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.5

Apply the product rule to $5 x$ .

$125 x^{3} - 3 (5^{2} x^{2}) y^{3} + 3 (5 x) {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.6

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

$125 x^{3} - 3 (25 x^{2}) y^{3} + 3 (5 x) {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.7

Multiply $25$ by $- 3$ .

$125 x^{3} - 75 x^{2} y^{3} + 3 (5 x) {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.8

Multiply $5$ by $3$ .

$125 x^{3} - 75 x^{2} y^{3} + 15 x {(- y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.9

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

$125 x^{3} - 75 x^{2} y^{3} + 15 x ({(- 1)}^{2} {(y^{3})}^{2}) + {(- y^{3})}^{3}$

Step 2.1.10

Rewrite using the commutative property of multiplication.

$125 x^{3} - 75 x^{2} y^{3} + 15 \cdot {(- 1)}^{2} x {(y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.11

Raise $- 1$ to the power of $2$ .

$125 x^{3} - 75 x^{2} y^{3} + 15 \cdot 1 x {(y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.12

Multiply $15$ by $1$ .

$125 x^{3} - 75 x^{2} y^{3} + 15 x {(y^{3})}^{2} + {(- y^{3})}^{3}$

Step 2.1.13

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

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Step 2.1.13.1

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

$125 x^{3} - 75 x^{2} y^{3} + 15 x y^{3 \cdot 2} + {(- y^{3})}^{3}$

Step 2.1.13.2

Multiply $3$ by $2$ .

$125 x^{3} - 75 x^{2} y^{3} + 15 x y^{6} + {(- y^{3})}^{3}$

Step 2.1.14

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

$125 x^{3} - 75 x^{2} y^{3} + 15 x y^{6} + {(- 1)}^{3} {(y^{3})}^{3}$

Step 2.1.15

Raise $- 1$ to the power of $3$ .

$125 x^{3} - 75 x^{2} y^{3} + 15 x y^{6} - {(y^{3})}^{3}$

Step 2.1.16

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

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Step 2.1.16.1

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

$125 x^{3} - 75 x^{2} y^{3} + 15 x y^{6} - y^{3 \cdot 3}$

Step 2.1.16.2

Multiply $3$ by $3$ .

$125 x^{3} - 75 x^{2} y^{3} + 15 x y^{6} - y^{9}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $125 x^{3}$ .

$- 75 x^{2} y^{3} + 15 x y^{6} - y^{9} + 125 x^{3}$

Step 2.2.2

Move $- 75 x^{2} y^{3}$ .

$15 x y^{6} - y^{9} - 75 x^{2} y^{3} + 125 x^{3}$

Step 2.2.3

Reorder $15 x y^{6}$ and $- y^{9}$ .

$- y^{9} + 15 x y^{6} - 75 x^{2} y^{3} + 125 x^{3}$