Precalculus Examples

$\frac{{(8 - 3 x)}^{\frac{1}{2}} + \frac{3}{2} x {(8 - 3 x)}^{- \frac{1}{2}}}{8 - 3 x}$

Step 1

Simplify the numerator.

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Step 1.1

Combine $\frac{3}{2}$ and $x$ .

$\frac{{(8 - 3 x)}^{\frac{1}{2}} + \frac{3 x}{2} {(8 - 3 x)}^{- \frac{1}{2}}}{8 - 3 x}$

Step 1.2

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

$\frac{{(8 - 3 x)}^{\frac{1}{2}} + \frac{3 x}{2} \cdot \frac{1}{{(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.3

Combine.

$\frac{{(8 - 3 x)}^{\frac{1}{2}} + \frac{3 x \cdot 1}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.4

Multiply $3$ by $1$ .

$\frac{{(8 - 3 x)}^{\frac{1}{2}} + \frac{3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.5

To write ${(8 - 3 x)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{2 {(8 - 3 x)}^{\frac{1}{2}}}{2 {(8 - 3 x)}^{\frac{1}{2}}}$ .

$\frac{{(8 - 3 x)}^{\frac{1}{2}} \cdot \frac{2 {(8 - 3 x)}^{\frac{1}{2}}}{2 {(8 - 3 x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.6

Combine ${(8 - 3 x)}^{\frac{1}{2}}$ and $\frac{2 {(8 - 3 x)}^{\frac{1}{2}}}{2 {(8 - 3 x)}^{\frac{1}{2}}}$ .

$\frac{\frac{{(8 - 3 x)}^{\frac{1}{2}} (2 {(8 - 3 x)}^{\frac{1}{2}})}{2 {(8 - 3 x)}^{\frac{1}{2}}} + \frac{3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.7

Combine the numerators over the common denominator.

$\frac{\frac{{(8 - 3 x)}^{\frac{1}{2}} (2 {(8 - 3 x)}^{\frac{1}{2}}) + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8

Rewrite $\frac{{(8 - 3 x)}^{\frac{1}{2}} (2 {(8 - 3 x)}^{\frac{1}{2}}) + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}$ in a factored form.

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Step 1.8.1

Rewrite using the commutative property of multiplication.

$\frac{\frac{2 {(8 - 3 x)}^{\frac{1}{2}} {(8 - 3 x)}^{\frac{1}{2}} + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.2

Multiply ${(8 - 3 x)}^{\frac{1}{2}}$ by ${(8 - 3 x)}^{\frac{1}{2}}$ by adding the exponents.

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Step 1.8.2.1

Move ${(8 - 3 x)}^{\frac{1}{2}}$ .

$\frac{\frac{2 ({(8 - 3 x)}^{\frac{1}{2}} {(8 - 3 x)}^{\frac{1}{2}}) + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.2.2

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

$\frac{\frac{2 {(8 - 3 x)}^{\frac{1}{2} + \frac{1}{2}} + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.2.3

Combine the numerators over the common denominator.

$\frac{\frac{2 {(8 - 3 x)}^{\frac{1 + 1}{2}} + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.2.4

Add $1$ and $1$ .

$\frac{\frac{2 {(8 - 3 x)}^{\frac{2}{2}} + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.2.5

Divide $2$ by $2$ .

$\frac{\frac{2 {(8 - 3 x)}^{1} + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.3

Simplify $2 {(8 - 3 x)}^{1}$ .

$\frac{\frac{2 (8 - 3 x) + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.4

Apply the distributive property.

$\frac{\frac{2 \cdot 8 + 2 (- 3 x) + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.5

Multiply $2$ by $8$ .

$\frac{\frac{16 + 2 (- 3 x) + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.6

Multiply $- 3$ by $2$ .

$\frac{\frac{16 - 6 x + 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 1.8.7

Add $- 6 x$ and $3 x$ .

$\frac{\frac{16 - 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}}{8 - 3 x}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{16 - 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}} \cdot \frac{1}{8 - 3 x}$

Step 3

Multiply $\frac{16 - 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}} \cdot \frac{1}{8 - 3 x}$ .

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Step 3.1

Multiply $\frac{16 - 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}}}$ by $\frac{1}{8 - 3 x}$ .

$\frac{16 - 3 x}{2 {(8 - 3 x)}^{\frac{1}{2}} (8 - 3 x)}$

Step 3.2

Raise $8 - 3 x$ to the power of $1$ .

$\frac{16 - 3 x}{2 ({(8 - 3 x)}^{1} {(8 - 3 x)}^{\frac{1}{2}})}$

Step 3.3

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

$\frac{16 - 3 x}{2 {(8 - 3 x)}^{1 + \frac{1}{2}}}$

Step 3.4

Write $1$ as a fraction with a common denominator.

$\frac{16 - 3 x}{2 {(8 - 3 x)}^{\frac{2}{2} + \frac{1}{2}}}$

Step 3.5

Combine the numerators over the common denominator.

$\frac{16 - 3 x}{2 {(8 - 3 x)}^{\frac{2 + 1}{2}}}$

Step 3.6

Add $2$ and $1$ .

$\frac{16 - 3 x}{2 {(8 - 3 x)}^{\frac{3}{2}}}$