Calculus Examples

$J = p \cdot \frac{D^{4} - {(D - 2 w)}^{4}}{32}$

Step 1

Rewrite the equation as $p \cdot \frac{D^{4} - {(D - 2 w)}^{4}}{32} = J$ .

$p \cdot \frac{D^{4} - {(D - 2 w)}^{4}}{32} = J$

Step 2

Simplify $p \cdot \frac{D^{4} - {(D - 2 w)}^{4}}{32}$ .

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Step 2.1

Simplify the numerator.

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Step 2.1.1

Rewrite $D^{4}$ as ${(D^{2})}^{2}$ .

$p \cdot \frac{{(D^{2})}^{2} - {(D - 2 w)}^{4}}{32} = J$

Step 2.1.2

Rewrite ${(D - 2 w)}^{4}$ as ${({(D - 2 w)}^{2})}^{2}$ .

$p \cdot \frac{{(D^{2})}^{2} - {({(D - 2 w)}^{2})}^{2}}{32} = J$

Step 2.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = D^{2}$ and $b = {(D - 2 w)}^{2}$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) (D^{2} - {(D - 2 w)}^{2})}{32} = J$

Step 2.1.4

Simplify.

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Step 2.1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = D$ and $b = D - 2 w$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) ((D + D - 2 w) (D - (D - 2 w)))}{32} = J$

Step 2.1.4.2

Simplify.

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Step 2.1.4.2.1

Add $D$ and $D$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) ((2 D - 2 w) (D - (D - 2 w)))}{32} = J$

Step 2.1.4.2.2

Factor $2$ out of $2 D - 2 w$ .

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Step 2.1.4.2.2.1

Factor $2$ out of $2 D$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) ((2 (D) - 2 w) (D - (D - 2 w)))}{32} = J$

Step 2.1.4.2.2.2

Factor $2$ out of $- 2 w$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) ((2 (D) + 2 (- w)) (D - (D - 2 w)))}{32} = J$

Step 2.1.4.2.2.3

Factor $2$ out of $2 (D) + 2 (- w)$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) (2 (D - w) (D - (D - 2 w)))}{32} = J$

Step 2.1.4.2.3

Apply the distributive property.

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) (2 (D - w) (D - D - (- 2 w)))}{32} = J$

Step 2.1.4.2.4

Multiply $- 2$ by $- 1$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) (2 (D - w) (D - D + 2 w))}{32} = J$

Step 2.1.4.2.5

Subtract $D$ from $D$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) (2 (D - w) (0 + 2 w))}{32} = J$

Step 2.1.4.2.6

Add $0$ and $2 w$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) (2 (D - w) \cdot 2 w)}{32} = J$

Step 2.1.4.2.7

Multiply $2$ by $2$ .

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) (4 (D - w) w)}{32} = J$

$p \cdot \frac{(D^{2} + {(D - 2 w)}^{2}) \cdot 4 (D - w) w}{32} = J$

Step 2.2

Simplify terms.

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Step 2.2.1

Cancel the common factor of $4$ and $32$ .

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Step 2.2.1.1

Factor $4$ out of $(D^{2} + {(D - 2 w)}^{2}) \cdot 4 (D - w) w$ .

$p \cdot \frac{4 (((D^{2} + {(D - 2 w)}^{2}) (D - w)) w)}{32} = J$

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $4$ out of $32$ .

$p \cdot \frac{4 (((D^{2} + {(D - 2 w)}^{2}) (D - w)) w)}{4 \cdot 8} = J$

Step 2.2.1.2.2

Cancel the common factor.

$p \cdot \frac{4 (((D^{2} + {(D - 2 w)}^{2}) (D - w)) w)}{4 \cdot 8} = J$

Step 2.2.1.2.3

Rewrite the expression.

$p \cdot \frac{((D^{2} + {(D - 2 w)}^{2}) (D - w)) w}{8} = J$

Step 2.2.2

Combine $p$ and $\frac{((D^{2} + {(D - 2 w)}^{2}) (D - w)) w}{8}$ .

$\frac{p (((D^{2} + {(D - 2 w)}^{2}) (D - w)) w)}{8} = J$

Step 2.2.3

Reorder factors in $\frac{p (D^{2} + {(D - 2 w)}^{2}) (D - w) w}{8}$ .

$\frac{p w (D^{2} + {(D - 2 w)}^{2}) (D - w)}{8} = J$