Precalculus Examples

$\sum_{k = 1}^{6} 6 k^{3}$

Step 1

Factor $6$ out of the summation.

$(6) \sum_{k = 1}^{6} k^{3}$

Step 2

The formula for the summation of a polynomial with degree $3$ is:

$\sum_{k = 1}^{n} k^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$

Step 3

Substitute the values into the formula and make sure to multiply by the front term.

$(6) (\frac{6^{2} {(6 + 1)}^{2}}{4})$

Step 4

Simplify.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Add $6$ and $1$ .

$6 \frac{6^{2} \cdot 7^{2}}{4}$

Step 4.1.2

Raise $6$ to the power of $2$ .

$6 \frac{36 \cdot 7^{2}}{4}$

Step 4.1.3

Raise $7$ to the power of $2$ .

$6 \frac{36 \cdot 49}{4}$

Step 4.2

Simplify terms.

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Step 4.2.1

Multiply $36$ by $49$ .

$6 (\frac{1764}{4})$

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Factor $2$ out of $6$ .

$2 (3) \frac{1764}{4}$

Step 4.2.2.2

Factor $2$ out of $4$ .

$2 \cdot 3 \frac{1764}{2 \cdot 2}$

Step 4.2.2.3

Cancel the common factor.

$2 \cdot 3 \frac{1764}{2 \cdot 2}$

Step 4.2.2.4

Rewrite the expression.

$3 (\frac{1764}{2})$

Step 4.2.3

Combine $3$ and $\frac{1764}{2}$ .

$\frac{3 \cdot 1764}{2}$

Step 4.2.4

Simplify the expression.

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Step 4.2.4.1

Multiply $3$ by $1764$ .

$\frac{5292}{2}$

Step 4.2.4.2

Divide $5292$ by $2$ .

$2646$