Calculus Examples

$\sum_{1}^{20} 8 {(7 x)}^{3}$

Step 1

Simplify the summation.

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Step 1.1

Apply the product rule to $7 x$ .

$8 (7^{3} x^{3})$

Step 1.2

Raise $7$ to the power of $3$ .

$8 (343 x^{3})$

Step 1.3

Multiply $343$ by $8$ .

$2744 x^{3}$

Step 1.4

Rewrite the summation.

$\sum_{x = 1}^{20} 2744 x^{3}$

Step 2

Factor $2744$ out of the summation.

$(2744) \sum_{x = 1}^{20} x^{3}$

Step 3

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 4

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

$(2744) (\frac{20^{2} {(20 + 1)}^{2}}{4})$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Add $20$ and $1$ .

$2744 \frac{20^{2} \cdot 21^{2}}{4}$

Step 5.1.2

Raise $20$ to the power of $2$ .

$2744 \frac{400 \cdot 21^{2}}{4}$

Step 5.1.3

Raise $21$ to the power of $2$ .

$2744 \frac{400 \cdot 441}{4}$

Step 5.2

Reduce the expression by cancelling the common factors.

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Step 5.2.1

Multiply $400$ by $441$ .

$2744 (\frac{176400}{4})$

Step 5.2.2

Cancel the common factor of $4$ .

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Step 5.2.2.1

Factor $4$ out of $2744$ .

$4 (686) \frac{176400}{4}$

Step 5.2.2.2

Cancel the common factor.

$4 \cdot 686 \frac{176400}{4}$

Step 5.2.2.3

Rewrite the expression.

$686 \cdot 176400$

Step 5.2.3

Multiply $686$ by $176400$ .

$121010400$