Simplify ((7 square root of x-x square root of 7)/(7x))/(x-7)

$\frac{(\frac{7 \sqrt{x} - x \sqrt{7}}{7 x})}{x - 7}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{7 \sqrt{x} - x \sqrt{7}}{7 x} \cdot \frac{1}{x - 7}$

Step 2

Simplify the numerator.

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

$\frac{7 x^{\frac{1}{2}} - x \sqrt{7}}{7 x} \cdot \frac{1}{x - 7}$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$ .

$\frac{7 x^{\frac{1}{2}} - x \cdot 7^{\frac{1}{2}}}{7 x} \cdot \frac{1}{x - 7}$

Factor $x^{\frac{1}{2}}$ out of $7 x^{\frac{1}{2}} - x \cdot 7^{\frac{1}{2}}$ .

Move $x$ .

$\frac{7 x^{\frac{1}{2}} - 1 \cdot 7^{\frac{1}{2}} x}{7 x} \cdot \frac{1}{x - 7}$

Factor $x^{\frac{1}{2}}$ out of $7 x^{\frac{1}{2}}$ .

$\frac{x^{\frac{1}{2}} \cdot 7 - 1 \cdot 7^{\frac{1}{2}} x}{7 x} \cdot \frac{1}{x - 7}$

Factor $x^{\frac{1}{2}}$ out of $- 1 \cdot 7^{\frac{1}{2}} x$ .

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

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

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

Rewrite $- 1 \cdot 7^{\frac{1}{2}}$ as $- 7^{\frac{1}{2}}$ .

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

Step 3

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

$\frac{7 - 7^{\frac{1}{2}} x^{\frac{1}{2}}}{7 x \cdot x^{- \frac{1}{2}}} \cdot \frac{1}{x - 7}$

Step 4

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

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

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

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

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

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

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

$\frac{7 - 7^{\frac{1}{2}} x^{\frac{1}{2}}}{7 x^{- \frac{1}{2} + 1}} \cdot \frac{1}{x - 7}$

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

$\frac{7 - 7^{\frac{1}{2}} x^{\frac{1}{2}}}{7 x^{- \frac{1}{2} + \frac{2}{2}}} \cdot \frac{1}{x - 7}$

Combine the numerators over the common denominator.

$\frac{7 - 7^{\frac{1}{2}} x^{\frac{1}{2}}}{7 x^{\frac{- 1 + 2}{2}}} \cdot \frac{1}{x - 7}$

Add $- 1$ and $2$ .

$\frac{7 - 7^{\frac{1}{2}} x^{\frac{1}{2}}}{7 x^{\frac{1}{2}}} \cdot \frac{1}{x - 7}$

Step 5

Multiply $\frac{7 - 7^{\frac{1}{2}} x^{\frac{1}{2}}}{7 x^{\frac{1}{2}}}$ by $\frac{1}{x - 7}$ .

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