Precalculus Examples

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

Step 1

Factor ${(x - 1)}^{\frac{3}{2}}$ out of ${(x - 1)}^{\frac{7}{2}} - {(x - 1)}^{\frac{3}{2}}$ .

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Step 1.1

Factor ${(x - 1)}^{\frac{3}{2}}$ out of ${(x - 1)}^{\frac{7}{2}}$ .

${(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{4}{2}} - {(x - 1)}^{\frac{3}{2}}$

Step 1.2

Factor ${(x - 1)}^{\frac{3}{2}}$ out of $- {(x - 1)}^{\frac{3}{2}}$ .

${(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{4}{2}} + {(x - 1)}^{\frac{3}{2}} \cdot - 1$

Step 1.3

Factor ${(x - 1)}^{\frac{3}{2}}$ out of ${(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{4}{2}} + {(x - 1)}^{\frac{3}{2}} \cdot - 1$ .

${(x - 1)}^{\frac{3}{2}} ({(x - 1)}^{\frac{4}{2}} - 1)$

Step 2

Rewrite ${(x - 1)}^{\frac{4}{2}}$ as ${({(x - 1)}^{\frac{2}{2}})}^{2}$ .

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

Step 3

Rewrite $1$ as $1^{2}$ .

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 1)}^{\frac{2}{2}}$ and $b = 1$ .

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

Step 5

Factor.

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Step 5.1

Simplify.

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Step 5.1.1

Divide $2$ by $2$ .

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

Step 5.1.2

Simplify.

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

Step 5.1.3

Add $- 1$ and $1$ .

${(x - 1)}^{\frac{3}{2}} ((x + 0) ({(x - 1)}^{\frac{2}{2}} - 1))$

Step 5.1.4

Add $x$ and $0$ .

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

Step 5.1.5

Rewrite ${(x - 1)}^{\frac{2}{2}}$ as ${({(x - 1)}^{\frac{1}{2}})}^{2}$ .

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

Step 5.1.6

Rewrite $1$ as $1^{2}$ .

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

Step 5.1.7

Factor.

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Step 5.1.7.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 1)}^{\frac{1}{2}}$ and $b = 1$ .

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

Step 5.1.7.2

Remove unnecessary parentheses.

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

Step 5.2

Remove unnecessary parentheses.

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