# Simplifying compound fraction: $\frac{3}{\sqrt{5}/5}$

I'm trying to simplify the following:

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

I know it is a very simple question but I am stuck. I followed through some instructions on Wolfram which suggests that I multiply the numerator by the reciprocal of the denominator.

The problem is I interpreted that as:

$$\frac{3}{\ \frac{\sqrt{5}}{5} \ } \times \frac{5}{\sqrt{5}},$$

Which I believe is:

$$\frac{15}{\ \frac{5}{5} \ } = \frac{15}{1}.$$

What am I doing wrong?

• You should multiply both numerator and denominator by that constant! Jul 23, 2013 at 9:50
• Or you can exchange the numerator and denominator of the whole denominator and move it to whole numerator! Jul 23, 2013 at 9:51
• I know this an old question, but if you had simply changed your √5/5 to a √5/√5, you would've got 3√5÷5/5 and gotten your answer. The whole point of multiplying a complex faction by a number to simplify it is to times it by 1 (√5/√5 in this case) because multiplying anything by 1 is the same thing. Sep 7, 2014 at 19:44
• "The problem is I interpreted that as:" .... remember that the numerator is 3. Jan 4, 2019 at 17:58

\begin{align*}\frac{3}{\frac{\sqrt{5}}{5}} &= 3 \cdot \frac{5}{\sqrt 5}\\ &= 3 \cdot \frac{5}{\sqrt 5} \cdot 1\\ &= 3 \cdot \frac{5}{\sqrt 5} \cdot \frac{\sqrt 5}{\sqrt 5}\\ &= 3 \cdot \frac{5\sqrt 5}{5}\\ &= 3\sqrt 5 \end{align*}

• I appreciate everyone's response but this answer is the most elaborate. Thanks blf, I see where I went wrong now!
– Sam
Jul 23, 2013 at 10:00

You multiplied the original fraction by $\dfrac5{\sqrt5}$, which is not $1$, so of course you changed the value. The correct course of action is to multiply by $1$ in the form

$$\frac{5/\sqrt5}{5/\sqrt5}$$

to get

$$\frac3{\frac{\sqrt5}5}=\frac3{\frac{\sqrt5}5}\cdot\frac{\frac5{\sqrt5}}{\frac5{\sqrt5}}=\frac{3\cdot\frac5{\sqrt5}}1=3\cdot\frac5{\sqrt5}=3\sqrt5\;,$$

since $\dfrac5{\sqrt5}=\sqrt5$.

More generally,

$$\frac{a}{b/c}=\frac{a}{\frac{b}c}\cdot\frac{\frac{c}b}{\frac{c}b}=\frac{a\cdot\frac{c}b}1=a\cdot\frac{c}b\;,$$

this is the basis for the invert and multiply rule for dividing by a fraction.

This means $$3\cdot \frac{5}{\sqrt{5}}=3\cdot\frac{(\sqrt{5})^2}{\sqrt{5}} =3\sqrt{5}$$ You're multiplying twice for the reciprocal of the denominator.

Another way to see it is multiplying numerator and denominator by the same number: $$\frac{3}{\frac{\sqrt{5}}{5}}=\frac{3\sqrt{5}}{\frac{\sqrt{5}}{5}\cdot\sqrt{5}} =\frac{3\sqrt{5}}{1}$$

Start with $$\frac{3}{\sqrt{5}/5}=\frac{15}{\sqrt{5}},$$

and then rationalize the denominator (multiply both numerator and denominator by $\sqrt{5}$) to get

$$\frac{15\sqrt{5}}{5}=3\sqrt{5}.$$

You have the following fraction to simplify: \begin{align} \frac{3}{\sqrt{5}/5} &=\frac{5\times 3}{\sqrt{5}} \\ &=\frac{15}{\sqrt{5}} \\ &=\sqrt{\bigg(\frac{15}{\sqrt{5}}\bigg)^2} \\ &=\sqrt{\frac{15^2}{\sqrt{5}^2}} \\ &=\sqrt{\frac{225}{5}} \\ &=\sqrt{45} \\ &= \sqrt{9\times 5} \\ &= \sqrt{9}\sqrt{5} \\ &= 3\sqrt{5} \\ \therefore \frac{15}{\sqrt{5}}\times \frac{5}{\sqrt{5}} &=\frac{15}{\sqrt{5}}\times \frac{\big(\frac{15}{\sqrt{5}}\big)}{3} \\ &=\frac{15}{\sqrt{5}}\times \frac{15}{\sqrt{5}}\times \frac{1}{3} \\ &=\bigg(\frac{15}{\sqrt{5}}\bigg)^2 \times \frac{1}{3} \\ &=45\times \frac{1}{3} \\ &=\frac{45}{3} \\ &=15 \end{align}

In the expression $$\frac{3}{\frac{\sqrt{5}}{5}}$$, to find the reciprocal of that expression's denominator, $$\frac{\sqrt{5}}{5}$$, simply swap its numerator and denominator.

Thus the reciprocal of $$\frac{\sqrt{5}}{5}$$ is $$\frac{5}{\sqrt{5}}$$, and $$\frac{3}{\frac{\sqrt{5}}{5}} = 3 \cdot \frac{5}{\sqrt{5}}$$

I think the result you have to remember is

$$\boxed{\dfrac 1{\sqrt{a}}=\dfrac{\sqrt{a}}a}$$

You will see this kind of manipulation very often in your studies, and it allows to get rid of square roots on denominator.

Here your expression is just $$\dfrac{3}{\frac{\sqrt{5}}{5}}=\dfrac{3}{\frac 1{\sqrt{5}}}=3\sqrt{5}$$

One thing that helps me organize my thoughts, is to convert both numerator and denuminator to fractions as follows.

\begin{align} \frac{3}{\frac{5}{\sqrt{5}}}&=\frac{\frac{3}{1}}{\frac{5}{\sqrt{5}}}=\frac{\frac{3}{1}}{\frac{5}{\sqrt{5}}}\cdot\frac{\frac{\sqrt{5}}{5}}{\frac{\sqrt{5}}{5}}=\frac{\frac{3}{1}\cdot\frac{\sqrt{5}}{5}}{\frac{5}{\sqrt{5}}\cdot\frac{\sqrt{5}}{5}}=\frac{\frac{3}{1}\cdot\frac{\sqrt{5}}{5}}{1}=\frac{3}{1}\cdot\frac{\sqrt{5}}{5}=\text{etc.}\\ \end{align}