Algebra Examples

${(a^{2} - 3 b^{3})}^{3}$

Step 1

Use the Binomial Theorem.

${(a^{2})}^{3} + 3 {(a^{2})}^{2} (- 3 b^{3}) + 3 a^{2} {(- 3 b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

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

$a^{2 \cdot 3} + 3 {(a^{2})}^{2} (- 3 b^{3}) + 3 a^{2} {(- 3 b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.1.2

Multiply $2$ by $3$ .

$a^{6} + 3 {(a^{2})}^{2} (- 3 b^{3}) + 3 a^{2} {(- 3 b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.2

Rewrite using the commutative property of multiplication.

$a^{6} + 3 \cdot - 3 {(a^{2})}^{2} b^{3} + 3 a^{2} {(- 3 b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.3

Multiply $3$ by $- 3$ .

$a^{6} - 9 {(a^{2})}^{2} b^{3} + 3 a^{2} {(- 3 b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.4

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

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Step 2.1.4.1

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

$a^{6} - 9 a^{2 \cdot 2} b^{3} + 3 a^{2} {(- 3 b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.4.2

Multiply $2$ by $2$ .

$a^{6} - 9 a^{4} b^{3} + 3 a^{2} {(- 3 b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.5

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

$a^{6} - 9 a^{4} b^{3} + 3 a^{2} ({(- 3)}^{2} {(b^{3})}^{2}) + {(- 3 b^{3})}^{3}$

Step 2.1.6

Rewrite using the commutative property of multiplication.

$a^{6} - 9 a^{4} b^{3} + 3 \cdot {(- 3)}^{2} a^{2} {(b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.7

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

$a^{6} - 9 a^{4} b^{3} + 3 \cdot 9 a^{2} {(b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.8

Multiply $3$ by $9$ .

$a^{6} - 9 a^{4} b^{3} + 27 a^{2} {(b^{3})}^{2} + {(- 3 b^{3})}^{3}$

Step 2.1.9

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

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Step 2.1.9.1

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

$a^{6} - 9 a^{4} b^{3} + 27 a^{2} b^{3 \cdot 2} + {(- 3 b^{3})}^{3}$

Step 2.1.9.2

Multiply $3$ by $2$ .

$a^{6} - 9 a^{4} b^{3} + 27 a^{2} b^{6} + {(- 3 b^{3})}^{3}$

Step 2.1.10

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

$a^{6} - 9 a^{4} b^{3} + 27 a^{2} b^{6} + {(- 3)}^{3} {(b^{3})}^{3}$

Step 2.1.11

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

$a^{6} - 9 a^{4} b^{3} + 27 a^{2} b^{6} - 27 {(b^{3})}^{3}$

Step 2.1.12

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

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Step 2.1.12.1

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

$a^{6} - 9 a^{4} b^{3} + 27 a^{2} b^{6} - 27 b^{3 \cdot 3}$

Step 2.1.12.2

Multiply $3$ by $3$ .

$a^{6} - 9 a^{4} b^{3} + 27 a^{2} b^{6} - 27 b^{9}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move $a^{6}$ .

$- 9 a^{4} b^{3} + 27 a^{2} b^{6} - 27 b^{9} + a^{6}$

Step 2.2.2

Move $- 9 a^{4} b^{3}$ .

$27 a^{2} b^{6} - 27 b^{9} - 9 a^{4} b^{3} + a^{6}$

Step 2.2.3

Reorder $27 a^{2} b^{6}$ and $- 27 b^{9}$ .

$- 27 b^{9} + 27 a^{2} b^{6} - 9 a^{4} b^{3} + a^{6}$