Algebra Examples

${(3 e^{2} - 5 f)}^{3}$

Step 1

Use the Binomial Theorem.

${(3 e^{2})}^{3} + 3 {(3 e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2

Simplify each term.

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Step 2.1

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

$3^{3} {(e^{2})}^{3} + 3 {(3 e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.2

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

$27 {(e^{2})}^{3} + 3 {(3 e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.3

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

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Step 2.3.1

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

$27 e^{2 \cdot 3} + 3 {(3 e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.3.2

Multiply $2$ by $3$ .

$27 e^{6} + 3 {(3 e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.4

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

$27 e^{6} + 3 (3^{2} {(e^{2})}^{2}) (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.5

Multiply $3$ by $3^{2}$ by adding the exponents.

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Step 2.5.1

Move $3^{2}$ .

$27 e^{6} + 3^{2} \cdot 3 {(e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.5.2

Multiply $3^{2}$ by $3$ .

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Step 2.5.2.1

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

$27 e^{6} + 3^{2} \cdot 3^{1} {(e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.5.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$27 e^{6} + 3^{2 + 1} {(e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.5.3

Add $2$ and $1$ .

$27 e^{6} + 3^{3} {(e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.6

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

$27 e^{6} + 27 {(e^{2})}^{2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.7

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

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Step 2.7.1

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

$27 e^{6} + 27 e^{2 \cdot 2} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.7.2

Multiply $2$ by $2$ .

$27 e^{6} + 27 e^{4} (- 5 f) + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.8

Multiply $- 5$ by $27$ .

$27 e^{6} - 135 e^{4} f + 3 (3 e^{2}) {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.9

Multiply $3$ by $3$ .

$27 e^{6} - 135 e^{4} f + 9 e^{2} {(- 5 f)}^{2} + {(- 5 f)}^{3}$

Step 2.10

Apply the product rule to $- 5 f$ .

$27 e^{6} - 135 e^{4} f + 9 e^{2} ({(- 5)}^{2} f^{2}) + {(- 5 f)}^{3}$

Step 2.11

Raise $- 5$ to the power of $2$ .

$27 e^{6} - 135 e^{4} f + 9 e^{2} (25 f^{2}) + {(- 5 f)}^{3}$

Step 2.12

Multiply $25$ by $9$ .

$27 e^{6} - 135 e^{4} f + 225 e^{2} f^{2} + {(- 5 f)}^{3}$

Step 2.13

Apply the product rule to $- 5 f$ .

$27 e^{6} - 135 e^{4} f + 225 e^{2} f^{2} + {(- 5)}^{3} f^{3}$

Step 2.14

Raise $- 5$ to the power of $3$ .

$27 e^{6} - 135 e^{4} f + 225 e^{2} f^{2} - 125 f^{3}$