Pre-Algebra Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

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

Step 2.1.1.2

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

$8 x^{2 \cdot 3} y^{3} + 3 {(2 x^{2} y)}^{2} (4 m) + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.3.2

Multiply $2$ by $3$ .

$8 x^{6} y^{3} + 3 {(2 x^{2} y)}^{2} (4 m) + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$8 x^{6} y^{3} + 3 \cdot 4 {(2 x^{2} y)}^{2} m + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.5

Multiply $3$ by $4$ .

$8 x^{6} y^{3} + 12 {(2 x^{2} y)}^{2} m + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.6

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

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Step 2.1.6.1

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

$8 x^{6} y^{3} + 12 ({(2 x^{2})}^{2} y^{2}) m + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.6.2

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

$8 x^{6} y^{3} + 12 (2^{2} {(x^{2})}^{2} y^{2}) m + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.7

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

$8 x^{6} y^{3} + 12 (4 {(x^{2})}^{2} y^{2}) m + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.8

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

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Step 2.1.8.1

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

$8 x^{6} y^{3} + 12 (4 x^{2 \cdot 2} y^{2}) m + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.8.2

Multiply $2$ by $2$ .

$8 x^{6} y^{3} + 12 (4 x^{4} y^{2}) m + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.9

Multiply $4$ by $12$ .

$8 x^{6} y^{3} + 48 (x^{4} y^{2}) m + 3 (2 x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.10

Multiply $2$ by $3$ .

$8 x^{6} y^{3} + 48 x^{4} y^{2} m + 6 (x^{2} y) {(4 m)}^{2} + {(4 m)}^{3}$

Step 2.1.11

Apply the product rule to $4 m$ .

$8 x^{6} y^{3} + 48 x^{4} y^{2} m + 6 x^{2} y (4^{2} m^{2}) + {(4 m)}^{3}$

Step 2.1.12

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

$8 x^{6} y^{3} + 48 x^{4} y^{2} m + 6 x^{2} y (16 m^{2}) + {(4 m)}^{3}$

Step 2.1.13

Multiply $16$ by $6$ .

$8 x^{6} y^{3} + 48 x^{4} y^{2} m + 96 x^{2} y m^{2} + {(4 m)}^{3}$

Step 2.1.14

Apply the product rule to $4 m$ .

$8 x^{6} y^{3} + 48 x^{4} y^{2} m + 96 x^{2} y m^{2} + 4^{3} m^{3}$

Step 2.1.15

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

$8 x^{6} y^{3} + 48 x^{4} y^{2} m + 96 x^{2} y m^{2} + 64 m^{3}$

Step 2.2

Reorder.

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Step 2.2.1

Move $y^{2}$ .

$8 x^{6} y^{3} + 48 x^{4} m y^{2} + 96 x^{2} y m^{2} + 64 m^{3}$

Step 2.2.2

Move $x^{4}$ .

$8 x^{6} y^{3} + 48 m x^{4} y^{2} + 96 x^{2} y m^{2} + 64 m^{3}$

Step 2.2.3

Move $y$ .

$8 x^{6} y^{3} + 48 m x^{4} y^{2} + 96 x^{2} m^{2} y + 64 m^{3}$

Step 2.2.4

Move $x^{2}$ .

$8 x^{6} y^{3} + 48 m x^{4} y^{2} + 96 m^{2} x^{2} y + 64 m^{3}$