Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Use the Binomial Theorem.

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

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the product rule to $2 x$ .

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

Step 1.2.2

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

$8 x^{3} + 3 {(2 x)}^{2} (- 3 y) + 3 (2 x) {(- 3 y)}^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.3

Rewrite using the commutative property of multiplication.

$8 x^{3} + 3 \cdot - 3 {(2 x)}^{2} y + 3 (2 x) {(- 3 y)}^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.4

Multiply $3$ by $- 3$ .

$8 x^{3} - 9 {(2 x)}^{2} y + 3 (2 x) {(- 3 y)}^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.5

Apply the product rule to $2 x$ .

$8 x^{3} - 9 (2^{2} x^{2}) y + 3 (2 x) {(- 3 y)}^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.6

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

$8 x^{3} - 9 (4 x^{2}) y + 3 (2 x) {(- 3 y)}^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.7

Multiply $4$ by $- 9$ .

$8 x^{3} - 36 x^{2} y + 3 (2 x) {(- 3 y)}^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.8

Multiply $2$ by $3$ .

$8 x^{3} - 36 x^{2} y + 6 x {(- 3 y)}^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.9

Apply the product rule to $- 3 y$ .

$8 x^{3} - 36 x^{2} y + 6 x ({(- 3)}^{2} y^{2}) + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.10

Rewrite using the commutative property of multiplication.

$8 x^{3} - 36 x^{2} y + 6 \cdot {(- 3)}^{2} x y^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.11

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

$8 x^{3} - 36 x^{2} y + 6 \cdot 9 x y^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.12

Multiply $6$ by $9$ .

$8 x^{3} - 36 x^{2} y + 54 x y^{2} + {(- 3 y)}^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.13

Apply the product rule to $- 3 y$ .

$8 x^{3} - 36 x^{2} y + 54 x y^{2} + {(- 3)}^{3} y^{3} - {(2 x + 3 y)}^{3}$

Step 1.2.14

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

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

Step 1.3

Use the Binomial Theorem.

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

Step 1.4

Simplify each term.

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Step 1.4.1

Apply the product rule to $2 x$ .

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

Step 1.4.2

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

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

Step 1.4.3

Rewrite using the commutative property of multiplication.

$8 x^{3} - 36 x^{2} y + 54 x y^{2} - 27 y^{3} - (8 x^{3} + 3 \cdot 3 {(2 x)}^{2} y + 3 (2 x) {(3 y)}^{2} + {(3 y)}^{3})$

Step 1.4.4

Multiply $3$ by $3$ .

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

Step 1.4.5

Apply the product rule to $2 x$ .

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

Step 1.4.6

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

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

Step 1.4.7

Multiply $4$ by $9$ .

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

Step 1.4.8

Multiply $2$ by $3$ .

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

Step 1.4.9

Apply the product rule to $3 y$ .

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

Step 1.4.10

Rewrite using the commutative property of multiplication.

$8 x^{3} - 36 x^{2} y + 54 x y^{2} - 27 y^{3} - (8 x^{3} + 36 x^{2} y + 6 \cdot 3^{2} x y^{2} + {(3 y)}^{3})$

Step 1.4.11

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

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

Step 1.4.12

Multiply $6$ by $9$ .

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

Step 1.4.13

Apply the product rule to $3 y$ .

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

Step 1.4.14

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

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

Step 1.5

Apply the distributive property.

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

Step 1.6

Simplify.

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Step 1.6.1

Multiply $8$ by $- 1$ .

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

Step 1.6.2

Multiply $36$ by $- 1$ .

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

Step 1.6.3

Multiply $54$ by $- 1$ .

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

Step 1.6.4

Multiply $27$ by $- 1$ .

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

Step 1.7

Remove parentheses.

$8 x^{3} - 36 x^{2} y + 54 x y^{2} - 27 y^{3} - 8 x^{3} - 36 x^{2} y - 54 x y^{2} - 27 y^{3}$

Step 2

Simplify by adding terms.

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Step 2.1

Combine the opposite terms in $8 x^{3} - 36 x^{2} y + 54 x y^{2} - 27 y^{3} - 8 x^{3} - 36 x^{2} y - 54 x y^{2} - 27 y^{3}$ .

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Step 2.1.1

Subtract $8 x^{3}$ from $8 x^{3}$ .

$- 36 x^{2} y + 54 x y^{2} - 27 y^{3} + 0 - 36 x^{2} y - 54 x y^{2} - 27 y^{3}$

Step 2.1.2

Add $- 36 x^{2} y + 54 x y^{2} - 27 y^{3}$ and $0$ .

$- 36 x^{2} y + 54 x y^{2} - 27 y^{3} - 36 x^{2} y - 54 x y^{2} - 27 y^{3}$

Step 2.1.3

Subtract $54 x y^{2}$ from $54 x y^{2}$ .

$- 36 x^{2} y - 27 y^{3} - 36 x^{2} y + 0 - 27 y^{3}$

Step 2.1.4

Add $- 36 x^{2} y - 27 y^{3} - 36 x^{2} y$ and $0$ .

$- 36 x^{2} y - 27 y^{3} - 36 x^{2} y - 27 y^{3}$

Step 2.2

Subtract $36 x^{2} y$ from $- 36 x^{2} y$ .

$- 27 y^{3} - 72 x^{2} y - 27 y^{3}$

Step 2.3

Subtract $27 y^{3}$ from $- 27 y^{3}$ .

$- 54 y^{3} - 72 x^{2} y$