Basic Math Examples

${(- a^{5} - b^{5})}^{3}$

Step 1

Use the Binomial Theorem.

${(- a^{5})}^{3} + 3 {(- a^{5})}^{2} (- b^{5}) + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $- a^{5}$ .

${(- 1)}^{3} {(a^{5})}^{3} + 3 {(- a^{5})}^{2} (- b^{5}) + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.2

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

$- {(a^{5})}^{3} + 3 {(- a^{5})}^{2} (- b^{5}) + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.3

Multiply the exponents in ${(a^{5})}^{3}$ .

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Step 2.3.1

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

$- a^{5 \cdot 3} + 3 {(- a^{5})}^{2} (- b^{5}) + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.3.2

Multiply $5$ by $3$ .

$- a^{15} + 3 {(- a^{5})}^{2} (- b^{5}) + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.4

Rewrite using the commutative property of multiplication.

$- a^{15} + 3 \cdot - 1 {(- a^{5})}^{2} b^{5} + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.5

Multiply $3$ by $- 1$ .

$- a^{15} - 3 {(- a^{5})}^{2} b^{5} + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.6

Apply the product rule to $- a^{5}$ .

$- a^{15} - 3 ({(- 1)}^{2} {(a^{5})}^{2}) b^{5} + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.7

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

$- a^{15} - 3 (1 {(a^{5})}^{2}) b^{5} + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.8

Multiply ${(a^{5})}^{2}$ by $1$ .

$- a^{15} - 3 {(a^{5})}^{2} b^{5} + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.9

Multiply the exponents in ${(a^{5})}^{2}$ .

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Step 2.9.1

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

$- a^{15} - 3 a^{5 \cdot 2} b^{5} + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.9.2

Multiply $5$ by $2$ .

$- a^{15} - 3 a^{10} b^{5} + 3 (- a^{5}) {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.10

Multiply $- 1$ by $3$ .

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} {(- b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.11

Apply the product rule to $- b^{5}$ .

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} ({(- 1)}^{2} {(b^{5})}^{2}) + {(- b^{5})}^{3}$

Step 2.12

Rewrite using the commutative property of multiplication.

$- a^{15} - 3 a^{10} b^{5} - 3 \cdot {(- 1)}^{2} a^{5} {(b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.13

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

$- a^{15} - 3 a^{10} b^{5} - 3 \cdot 1 a^{5} {(b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.14

Multiply $- 3$ by $1$ .

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} {(b^{5})}^{2} + {(- b^{5})}^{3}$

Step 2.15

Multiply the exponents in ${(b^{5})}^{2}$ .

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Step 2.15.1

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

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} b^{5 \cdot 2} + {(- b^{5})}^{3}$

Step 2.15.2

Multiply $5$ by $2$ .

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} b^{10} + {(- b^{5})}^{3}$

Step 2.16

Apply the product rule to $- b^{5}$ .

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} b^{10} + {(- 1)}^{3} {(b^{5})}^{3}$

Step 2.17

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

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} b^{10} - {(b^{5})}^{3}$

Step 2.18

Multiply the exponents in ${(b^{5})}^{3}$ .

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Step 2.18.1

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

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} b^{10} - b^{5 \cdot 3}$

Step 2.18.2

Multiply $5$ by $3$ .

$- a^{15} - 3 a^{10} b^{5} - 3 a^{5} b^{10} - b^{15}$