Precalculus Examples

${(x - 7)}^{\frac{7}{2}} - 49 {(x - 7)}^{\frac{3}{2}}$

Step 1

Factor ${(x - 7)}^{\frac{3}{2}}$ out of ${(x - 7)}^{\frac{7}{2}} - 49 {(x - 7)}^{\frac{3}{2}}$ .

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Step 1.1

Factor ${(x - 7)}^{\frac{3}{2}}$ out of ${(x - 7)}^{\frac{7}{2}}$ .

${(x - 7)}^{\frac{3}{2}} {(x - 7)}^{\frac{4}{2}} - 49 {(x - 7)}^{\frac{3}{2}}$

Step 1.2

Factor ${(x - 7)}^{\frac{3}{2}}$ out of $- 49 {(x - 7)}^{\frac{3}{2}}$ .

${(x - 7)}^{\frac{3}{2}} {(x - 7)}^{\frac{4}{2}} + {(x - 7)}^{\frac{3}{2}} \cdot - 49$

Step 1.3

Factor ${(x - 7)}^{\frac{3}{2}}$ out of ${(x - 7)}^{\frac{3}{2}} {(x - 7)}^{\frac{4}{2}} + {(x - 7)}^{\frac{3}{2}} \cdot - 49$ .

${(x - 7)}^{\frac{3}{2}} ({(x - 7)}^{\frac{4}{2}} - 49)$

Step 2

Rewrite ${(x - 7)}^{\frac{4}{2}}$ as ${({(x - 7)}^{\frac{2}{2}})}^{2}$ .

${(x - 7)}^{\frac{3}{2}} ({({(x - 7)}^{\frac{2}{2}})}^{2} - 49)$

Step 3

Rewrite $49$ as $7^{2}$ .

${(x - 7)}^{\frac{3}{2}} ({({(x - 7)}^{\frac{2}{2}})}^{2} - 7^{2})$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 7)}^{\frac{2}{2}}$ and $b = 7$ .

${(x - 7)}^{\frac{3}{2}} (({(x - 7)}^{\frac{2}{2}} + 7) ({(x - 7)}^{\frac{2}{2}} - 7))$

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Divide $2$ by $2$ .

${(x - 7)}^{\frac{3}{2}} (({(x - 7)}^{1} + 7) ({(x - 7)}^{\frac{2}{2}} - 7))$

Step 5.1.2

Simplify.

${(x - 7)}^{\frac{3}{2}} ((x - 7 + 7) ({(x - 7)}^{\frac{2}{2}} - 7))$

Step 5.1.3

Add $- 7$ and $7$ .

${(x - 7)}^{\frac{3}{2}} ((x + 0) ({(x - 7)}^{\frac{2}{2}} - 7))$

Step 5.1.4

Add $x$ and $0$ .

${(x - 7)}^{\frac{3}{2}} (x ({(x - 7)}^{\frac{2}{2}} - 7))$

Step 5.1.5

Divide $2$ by $2$ .

${(x - 7)}^{\frac{3}{2}} (x ({(x - 7)}^{1} - 7))$

Step 5.1.6

Simplify.

${(x - 7)}^{\frac{3}{2}} (x (x - 7 - 7))$

Step 5.1.7

Subtract $7$ from $- 7$ .

${(x - 7)}^{\frac{3}{2}} (x (x - 14))$

Step 5.2

Remove unnecessary parentheses.

${(x - 7)}^{\frac{3}{2}} x (x - 14)$