Mfrhtyj

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?

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{5sqrt 5}{5}\
&= 3sqrt 5
end{align*}

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

$$frac3{frac{sqrt5}5}=frac3{frac{sqrt5}5}cdotfrac{frac5{sqrt5}}{frac5{sqrt5}}=frac{3cdotfrac5{sqrt5}}1=3cdotfrac5{sqrt5}=3sqrt5;,$$

since $dfrac5{sqrt5}=sqrt5$.

More generally,

$$frac{a}{b/c}=frac{a}{frac{b}c}cdotfrac{frac{c}b}{frac{c}b}=frac{acdotfrac{c}b}1=acdotfrac{c}b;,$$

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

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

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

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

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

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

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

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