Exercises: Making Denominators Rational

With these exercises, you will learn how to make the denominator of a fraction with square roots rational.

1
Make the denominator rational. Simplify as far as possible.
$\frac{1}{\sqrt{2}}$

For this task you need the following basic knowledge: Square Roots
Expand with the denominator.
$\frac{1}{\sqrt{2}}$ $=$ $\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}$
$=$ $\frac{\sqrt{2}}{{(\sqrt{2})}^{2}}$
↓
The square root cancels the square.
$=$ $\frac{\sqrt{2}}{2}$

$\frac{5}{2 \sqrt{3}}$

For this task you need the following basic knowledge: Square Roots
Expand with $\sqrt{3}$ .
$\frac{5}{2 \sqrt{3}}$ $=$ $\frac{5 \sqrt{3}}{2 \cdot 3}$
$=$ $\frac{5}{2 \cdot 3} \cdot \sqrt{3}$
↓
Conclude.
$=$ $\frac{5}{6} \sqrt{3}$

$\frac{25}{\sqrt{125}}$

For this task you need the following basic knowledge: Square Roots
$\frac{25}{\sqrt{125}}$ $=$ $\frac{25}{\sqrt{25 \cdot 5}}$
↓
Take the square root of 25.
$=$ $\frac{25}{5 \sqrt{5}}$
↓
Expand with $\sqrt{5}$ .
$=$ $\frac{25 \cdot \sqrt{5}}{5 \cdot \sqrt{5} \cdot \sqrt{5}}$
$=$ $\frac{25 \cdot \sqrt{5}}{5 \cdot 5}$
↓
Shorten.
$=$ $\sqrt{5}$

$\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x y}}$

For this task you need the following basic knowledge: Square Roots
Expand with the denominator.
$\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x y}}$ $=$ $\frac{(\sqrt{x} - \sqrt{y}) \cdot \sqrt{x y}}{\sqrt{x y} \cdot \sqrt{x y}}$
$=$ $\frac{\sqrt{x} \cdot \sqrt{x y} - \sqrt{y} \cdot \sqrt{x y}}{{\sqrt{x y}}^{2}}$
↓
The square root cancels the square.
$=$ $\frac{{\sqrt{x}}^{2} \cdot \sqrt{y} - {\sqrt{y}}^{2} \cdot \sqrt{x}}{x y}$
$=$ $\frac{x \sqrt{y} - y \sqrt{x}}{x y}$
$=$ $\frac{x \sqrt{y}}{x y} - \frac{y \sqrt{x}}{x y}$
↓
Shorten.
$=$ $\frac{\sqrt{y}}{y} - \frac{\sqrt{x}}{x}$
2
Make the denominator rational:
$\frac{1}{\sqrt{2}}$

For this task you need the following basic knowledge: Square Roots
$\frac{1}{\sqrt{2}}$ $=$ $\frac{1}{\sqrt{2}}$
↓
Expand the fraction by $\sqrt{2}$ .
$=$ $\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}$
$=$ $\frac{\sqrt{2}}{2}$

$\frac{\sqrt{2} - \sqrt{125}}{\sqrt{5}}$

For this task you need the following basic knowledge: Square Roots
$\frac{\sqrt{2} - \sqrt{125}}{\sqrt{5}}$ $=$ $\frac{\sqrt{2} - \sqrt{125}}{\sqrt{5}}$
↓
Expand the fraction by $\sqrt{5}$ .
$=$ $\frac{(\sqrt{2} - \sqrt{125}) \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}$
↓
Multiply.
$=$ $\frac{\sqrt{10} - \sqrt{625}}{5}$
↓
Take the square root.
$=$ $\frac{\sqrt{10} - 25}{5}$
$=$ $\frac{\sqrt{10}}{5} - 5$

$\frac{1}{\sqrt{2}}$	$=$	$\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}$
	$=$	$\frac{\sqrt{2}}{{(\sqrt{2})}^{2}}$
	↓	The square root cancels the square.
	$=$	$\frac{\sqrt{2}}{2}$

$\frac{5}{2 \sqrt{3}}$	$=$	$\frac{5 \sqrt{3}}{2 \cdot 3}$
	$=$	$\frac{5}{2 \cdot 3} \cdot \sqrt{3}$
	↓	Conclude.
	$=$	$\frac{5}{6} \sqrt{3}$

$\frac{25}{\sqrt{125}}$	$=$	$\frac{25}{\sqrt{25 \cdot 5}}$
	↓	Take the square root of 25.
	$=$	$\frac{25}{5 \sqrt{5}}$
	↓	Expand with $\sqrt{5}$ .
	$=$	$\frac{25 \cdot \sqrt{5}}{5 \cdot \sqrt{5} \cdot \sqrt{5}}$
	$=$	$\frac{25 \cdot \sqrt{5}}{5 \cdot 5}$
	↓	Shorten.
	$=$	$\sqrt{5}$

$\frac{\sqrt{x} - \sqrt{y}}{\sqrt{x y}}$	$=$	$\frac{(\sqrt{x} - \sqrt{y}) \cdot \sqrt{x y}}{\sqrt{x y} \cdot \sqrt{x y}}$
	$=$	$\frac{\sqrt{x} \cdot \sqrt{x y} - \sqrt{y} \cdot \sqrt{x y}}{{\sqrt{x y}}^{2}}$
	↓	The square root cancels the square.
	$=$	$\frac{{\sqrt{x}}^{2} \cdot \sqrt{y} - {\sqrt{y}}^{2} \cdot \sqrt{x}}{x y}$
	$=$	$\frac{x \sqrt{y} - y \sqrt{x}}{x y}$
	$=$	$\frac{x \sqrt{y}}{x y} - \frac{y \sqrt{x}}{x y}$
	↓	Shorten.
	$=$	$\frac{\sqrt{y}}{y} - \frac{\sqrt{x}}{x}$

$\frac{1}{\sqrt{2}}$	$=$	$\frac{1}{\sqrt{2}}$
	↓	Expand the fraction by $\sqrt{2}$ .
	$=$	$\frac{1 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}}$
	$=$	$\frac{\sqrt{2}}{2}$

$\frac{\sqrt{2} - \sqrt{125}}{\sqrt{5}}$	$=$	$\frac{\sqrt{2} - \sqrt{125}}{\sqrt{5}}$
	↓	Expand the fraction by $\sqrt{5}$ .
	$=$	$\frac{(\sqrt{2} - \sqrt{125}) \cdot \sqrt{5}}{\sqrt{5} \cdot \sqrt{5}}$
	↓	Multiply.
	$=$	$\frac{\sqrt{10} - \sqrt{625}}{5}$
	↓	Take the square root.
	$=$	$\frac{\sqrt{10} - 25}{5}$
	$=$	$\frac{\sqrt{10}}{5} - 5$