Exercises: Making Denominators Rational

…

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

Make the denominator rational. Simplify as far as possible.

$\frac1{\sqrt2}$

Für diese Aufgabe benötigst Du folgendes Grundwissen: Square Roots

Expand with the denominator.

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

Do you have a question?

$\frac5{2\sqrt3}$

Für diese Aufgabe benötigst Du folgendes Grundwissen: Square Roots

Expand with $\sqrt{3}$ .

$\displaystyle \frac{5}{2\sqrt{3}}$	$=$	$\displaystyle \frac{5\sqrt{3}}{2\cdot3}$
	$=$	$\displaystyle \frac{5}{2\cdot3}\cdot\sqrt{3}$
	↓	Conclude.
	$=$	$\displaystyle \frac{5}{6}\sqrt{3}$

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

Für diese Aufgabe benötigst Du folgendes Grundwissen: Square Roots

$\displaystyle \frac{25}{\sqrt{125}}$	$=$	$\displaystyle \frac{25}{\sqrt{25\cdot5}}$
	↓	Take the square root of 25.
	$=$	$\displaystyle \frac{25}{5\sqrt{5}}$
	↓	Expand with $\sqrt{5}$ .
	$=$	$\displaystyle \frac{25\cdot\sqrt{5}}{5\cdot\sqrt{5}\cdot\sqrt{5}}$
	$=$	$\displaystyle \frac{25\cdot\sqrt{5}}{5\cdot5}$
	↓	Shorten.
	$=$	$\displaystyle \sqrt{5}$

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

Für diese Aufgabe benötigst Du folgendes Grundwissen: Square Roots

Expand with the denominator.

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

Make the denominator rational:

$\frac{1}{\sqrt{2}}$

Für diese Aufgabe benötigst Du folgendes Grundwissen: Square Roots

$\displaystyle \frac{1}{\sqrt{2}}$	$=$	$\displaystyle \frac{1}{\sqrt{2}}$
	↓	Expand the fraction by $\sqrt2$ .
	$=$	$\displaystyle \frac{1\cdot\sqrt{2}}{\sqrt{2}\cdot\sqrt{2}}$
	$=$	$\displaystyle \frac{\sqrt{2}}{2}$

$\frac{\sqrt2-\sqrt{125}}{\sqrt5}$

Für diese Aufgabe benötigst Du folgendes Grundwissen: Square Roots

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

This content is licensed under
CC BY-SA 4.0

→ Info