Simplify ((6 square root of x-x square root of 6)/(6x))/(x-6)

$\frac{(\frac{6 \sqrt{x} - x \sqrt{6}}{6 x})}{x - 6}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{6 \sqrt{x} - x \sqrt{6}}{6 x} \cdot \frac{1}{x - 6}$

Step 2

Simplify the numerator.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{6 x^{\frac{1}{2}} - x \sqrt{6}}{6 x} \cdot \frac{1}{x - 6}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$\frac{6 x^{\frac{1}{2}} - x \cdot 6^{\frac{1}{2}}}{6 x} \cdot \frac{1}{x - 6}$

Factor $x^{\frac{1}{2}}$ out of $6 x^{\frac{1}{2}} - x \cdot 6^{\frac{1}{2}}$ .

Move $x$ .

$\frac{6 x^{\frac{1}{2}} - 1 \cdot 6^{\frac{1}{2}} x}{6 x} \cdot \frac{1}{x - 6}$

Factor $x^{\frac{1}{2}}$ out of $6 x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} \cdot 6 - 1 \cdot 6^{\frac{1}{2}} x}{6 x} \cdot \frac{1}{x - 6}$

Factor $x^{\frac{1}{2}}$ out of $- 1 \cdot 6^{\frac{1}{2}} x$ .

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

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

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

Rewrite $- 1 \cdot 6^{\frac{1}{2}}$ as $- 6^{\frac{1}{2}}$ .

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

Step 3

Move $x^{\frac{1}{2}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\frac{6 - 6^{\frac{1}{2}} x^{\frac{1}{2}}}{6 x \cdot x^{- \frac{1}{2}}} \cdot \frac{1}{x - 6}$

Step 4

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

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

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

Multiply $x^{- \frac{1}{2}}$ by $x$ .

Raise $x$ to the power of $1$ .

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

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

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

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

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

Combine the numerators over the common denominator.

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

Add $- 1$ and $2$ .

$\frac{6 - 6^{\frac{1}{2}} x^{\frac{1}{2}}}{6 x^{\frac{1}{2}}} \cdot \frac{1}{x - 6}$

Step 5

Multiply $\frac{6 - 6^{\frac{1}{2}} x^{\frac{1}{2}}}{6 x^{\frac{1}{2}}}$ by $\frac{1}{x - 6}$ .

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