Algebra Examples

$(3 e^{2} - 5 e + 7) (6 e + 1)$

Step 1

Expand $(3 e^{2} - 5 e + 7) (6 e + 1)$ by multiplying each term in the first expression by each term in the second expression.

$3 e^{2} (6 e) + 3 e^{2} \cdot 1 - 5 e (6 e) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Move $e$ .

$3 (e e^{2}) \cdot 6 + 3 e^{2} \cdot 1 - 5 e (6 e) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.1.2

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

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Step 2.1.1.2.1

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

$3 (e^{1} e^{2}) \cdot 6 + 3 e^{2} \cdot 1 - 5 e (6 e) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.1.2.2

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

$3 e^{1 + 2} \cdot 6 + 3 e^{2} \cdot 1 - 5 e (6 e) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.1.3

Add $1$ and $2$ .

$3 e^{3} \cdot 6 + 3 e^{2} \cdot 1 - 5 e (6 e) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.2

Multiply $6$ by $3$ .

$18 e^{3} + 3 e^{2} \cdot 1 - 5 e (6 e) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.3

Multiply $3$ by $1$ .

$18 e^{3} + 3 e^{2} - 5 e (6 e) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.4

Multiply $- 5 e (6 e)$ .

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Step 2.1.4.1

Multiply $6$ by $- 5$ .

$18 e^{3} + 3 e^{2} - 30 e e - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.4.2

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

$18 e^{3} + 3 e^{2} - 30 (e^{1} e) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.4.3

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

$18 e^{3} + 3 e^{2} - 30 (e^{1} e^{1}) - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.4.4

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

$18 e^{3} + 3 e^{2} - 30 e^{1 + 1} - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.4.5

Add $1$ and $1$ .

$18 e^{3} + 3 e^{2} - 30 e^{2} - 5 e \cdot 1 + 7 (6 e) + 7 \cdot 1$

Step 2.1.5

Multiply $- 5$ by $1$ .

$18 e^{3} + 3 e^{2} - 30 e^{2} - 5 e + 7 (6 e) + 7 \cdot 1$

Step 2.1.6

Multiply $6$ by $7$ .

$18 e^{3} + 3 e^{2} - 30 e^{2} - 5 e + 42 e + 7 \cdot 1$

Step 2.1.7

Multiply $7$ by $1$ .

$18 e^{3} + 3 e^{2} - 30 e^{2} - 5 e + 42 e + 7$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $30 e^{2}$ from $3 e^{2}$ .

$18 e^{3} - 27 e^{2} - 5 e + 42 e + 7$

Step 2.2.2

Add $- 5 e$ and $42 e$ .

$18 e^{3} - 27 e^{2} + 37 e + 7$

Step 3

The result can be shown in multiple forms.

Exact Form:

$18 e^{3} - 27 e^{2} + 37 e + 7$

Decimal Form:

$269.61157759 \dots$