Algebra Examples

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

Step 1

Use the Binomial Theorem.

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

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 $3 y^{2}$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

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Step 2.1.3.1

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

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

Step 2.1.3.2

Multiply $2$ by $3$ .

$27 y^{6} + 3 {(3 y^{2})}^{2} (2 x^{2}) + 3 (3 y^{2}) {(2 x^{2})}^{2} + {(2 x^{2})}^{3}$

Step 2.1.4

Rewrite using the commutative property of multiplication.

$27 y^{6} + 3 \cdot 2 {(3 y^{2})}^{2} x^{2} + 3 (3 y^{2}) {(2 x^{2})}^{2} + {(2 x^{2})}^{3}$

Step 2.1.5

Multiply $3$ by $2$ .

$27 y^{6} + 6 {(3 y^{2})}^{2} x^{2} + 3 (3 y^{2}) {(2 x^{2})}^{2} + {(2 x^{2})}^{3}$

Step 2.1.6

Apply the product rule to $3 y^{2}$ .

$27 y^{6} + 6 (3^{2} {(y^{2})}^{2}) x^{2} + 3 (3 y^{2}) {(2 x^{2})}^{2} + {(2 x^{2})}^{3}$

Step 2.1.7

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

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

Step 2.1.8

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

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Step 2.1.8.1

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

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

Step 2.1.8.2

Multiply $2$ by $2$ .

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

Step 2.1.9

Multiply $9$ by $6$ .

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

Step 2.1.10

Multiply $3$ by $3$ .

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

Step 2.1.11

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

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

Step 2.1.12

Rewrite using the commutative property of multiplication.

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

Step 2.1.13

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

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

Step 2.1.14

Multiply $9$ by $4$ .

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

Step 2.1.15

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

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Step 2.1.15.1

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

$27 y^{6} + 54 y^{4} x^{2} + 36 y^{2} x^{2 \cdot 2} + {(2 x^{2})}^{3}$

Step 2.1.15.2

Multiply $2$ by $2$ .

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

Step 2.1.16

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

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

Step 2.1.17

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

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

Step 2.1.18

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

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Step 2.1.18.1

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

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

Step 2.1.18.2

Multiply $2$ by $3$ .

$27 y^{6} + 54 y^{4} x^{2} + 36 y^{2} x^{4} + 8 x^{6}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $y^{4}$ .

$27 y^{6} + 54 x^{2} y^{4} + 36 y^{2} x^{4} + 8 x^{6}$

Step 2.2.2

Move $y^{2}$ .

$27 y^{6} + 54 x^{2} y^{4} + 36 x^{4} y^{2} + 8 x^{6}$

Step 2.2.3

Move $27 y^{6}$ .

$54 x^{2} y^{4} + 36 x^{4} y^{2} + 8 x^{6} + 27 y^{6}$

Step 2.2.4

Move $54 x^{2} y^{4}$ .

$36 x^{4} y^{2} + 8 x^{6} + 54 x^{2} y^{4} + 27 y^{6}$

Step 2.2.5

Reorder $36 x^{4} y^{2}$ and $8 x^{6}$ .

$8 x^{6} + 36 x^{4} y^{2} + 54 x^{2} y^{4} + 27 y^{6}$