Precalculus Examples

$3 {(x - 1)}^{\frac{3}{2}} + 6 {(x - 1)}^{\frac{1}{2}} - 9 {(x - 1)}^{- \frac{1}{2}}$

Step 1

Simplify each term.

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Step 1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$3 {(x - 1)}^{\frac{3}{2}} + 6 {(x - 1)}^{\frac{1}{2}} - 9 \frac{1}{{(x - 1)}^{\frac{1}{2}}}$

Step 1.2

Combine $- 9$ and $\frac{1}{{(x - 1)}^{\frac{1}{2}}}$ .

$3 {(x - 1)}^{\frac{3}{2}} + 6 {(x - 1)}^{\frac{1}{2}} + \frac{- 9}{{(x - 1)}^{\frac{1}{2}}}$

Step 1.3

Move the negative in front of the fraction.

$3 {(x - 1)}^{\frac{3}{2}} + 6 {(x - 1)}^{\frac{1}{2}} - \frac{9}{{(x - 1)}^{\frac{1}{2}}}$

Step 2

To write $3 {(x - 1)}^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x - 1)}^{\frac{1}{2}}}{{(x - 1)}^{\frac{1}{2}}}$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 {(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{1}{2}}}{{(x - 1)}^{\frac{1}{2}}} - \frac{9}{{(x - 1)}^{\frac{1}{2}}}$

Step 3

Combine the numerators over the common denominator.

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 {(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{1}{2}} - 9}{{(x - 1)}^{\frac{1}{2}}}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $3$ out of $3 {(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{1}{2}} - 9$ .

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Step 4.1.1

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

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 ({(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{1}{2}}) - 9}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.1.2

Factor $3$ out of $- 9$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 ({(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{1}{2}}) + 3 \cdot - 3}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.1.3

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

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 ({(x - 1)}^{\frac{3}{2}} {(x - 1)}^{\frac{1}{2}} - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.2

Multiply ${(x - 1)}^{\frac{3}{2}}$ by ${(x - 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 ({(x - 1)}^{\frac{3}{2} + \frac{1}{2}} - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.2.2

Combine the numerators over the common denominator.

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 ({(x - 1)}^{\frac{3 + 1}{2}} - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.2.3

Add $3$ and $1$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 ({(x - 1)}^{\frac{4}{2}} - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.2.4

Divide $4$ by $2$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 ({(x - 1)}^{2} - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.3

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

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 ((x - 1) (x - 1) - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.4

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 4.4.1

Apply the distributive property.

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x (x - 1) - 1 (x - 1) - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.4.2

Apply the distributive property.

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x \cdot x + x \cdot - 1 - 1 (x - 1) - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.4.3

Apply the distributive property.

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.5

Simplify and combine like terms.

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Step 4.5.1

Simplify each term.

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Step 4.5.1.1

Multiply $x$ by $x$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.5.1.2

Move $- 1$ to the left of $x$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.5.1.3

Rewrite $- 1 x$ as $- x$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x^{2} - x - 1 x - 1 \cdot - 1 - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.5.1.4

Rewrite $- 1 x$ as $- x$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x^{2} - x - x - 1 \cdot - 1 - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.5.1.5

Multiply $- 1$ by $- 1$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x^{2} - x - x + 1 - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.5.2

Subtract $x$ from $- x$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x^{2} - 2 x + 1 - 3)}{{(x - 1)}^{\frac{1}{2}}}$

Step 4.6

Subtract $3$ from $1$ .

$6 {(x - 1)}^{\frac{1}{2}} + \frac{3 (x^{2} - 2 x - 2)}{{(x - 1)}^{\frac{1}{2}}}$

Step 5

To write $6 {(x - 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x - 1)}^{\frac{1}{2}}}{{(x - 1)}^{\frac{1}{2}}}$ .

$\frac{6 {(x - 1)}^{\frac{1}{2}} {(x - 1)}^{\frac{1}{2}}}{{(x - 1)}^{\frac{1}{2}}} + \frac{3 (x^{2} - 2 x - 2)}{{(x - 1)}^{\frac{1}{2}}}$

Step 6

Combine the numerators over the common denominator.

$\frac{6 {(x - 1)}^{\frac{1}{2}} {(x - 1)}^{\frac{1}{2}} + 3 (x^{2} - 2 x - 2)}{{(x - 1)}^{\frac{1}{2}}}$

Step 7

Simplify the numerator.

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Step 7.1

Factor $3$ out of $6 {(x - 1)}^{\frac{1}{2}} {(x - 1)}^{\frac{1}{2}} + 3 (x^{2} - 2 x - 2)$ .

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Step 7.1.1

Factor $3$ out of $6 {(x - 1)}^{\frac{1}{2}} {(x - 1)}^{\frac{1}{2}}$ .

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

Step 7.1.2

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

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

Step 7.2

Multiply ${(x - 1)}^{\frac{1}{2}}$ by ${(x - 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 7.2.1

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

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

Step 7.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 7.2.3

Combine the numerators over the common denominator.

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

Step 7.2.4

Add $1$ and $1$ .

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

Step 7.2.5

Divide $2$ by $2$ .

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

Step 7.3

Simplify $2 {(x - 1)}^{1}$ .

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

Step 7.4

Apply the distributive property.

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

Step 7.5

Multiply $2$ by $- 1$ .

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

Step 7.6

Subtract $2 x$ from $2 x$ .

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

Step 7.7

Add $- 2$ and $0$ .

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

Step 7.8

Subtract $2$ from $- 2$ .

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

Step 7.9

Rewrite $x^{2} - 4$ in a factored form.

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Step 7.9.1

Rewrite $4$ as $2^{2}$ .

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

Step 7.9.2

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

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

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