5
$\begingroup$

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?

$\endgroup$
  • 1
    $\begingroup$ You should multiply both numerator and denominator by that constant! $\endgroup$ – Mahdi Khosravi Jul 23 '13 at 9:50
  • $\begingroup$ Or you can exchange the numerator and denominator of the whole denominator and move it to whole numerator! $\endgroup$ – Mahdi Khosravi Jul 23 '13 at 9:51
  • $\begingroup$ 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. $\endgroup$ – Gᴇᴏᴍᴇᴛᴇʀ Sep 7 '14 at 19:44
  • $\begingroup$ "The problem is I interpreted that as:" .... remember that the numerator is 3. $\endgroup$ – John Joy Jan 4 at 17:58
5
$\begingroup$

\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*}

$\endgroup$
  • $\begingroup$ I appreciate everyone's response but this answer is the most elaborate. Thanks blf, I see where I went wrong now! $\endgroup$ – Sam Jul 23 '13 at 10:00
5
$\begingroup$

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.

$\endgroup$
4
$\begingroup$

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: $$ \frac{3}{\frac{\sqrt{5}}{5}}=\frac{3\sqrt{5}}{\frac{\sqrt{5}}{5}\cdot\sqrt{5}} =\frac{3\sqrt{5}}{1} $$

$\endgroup$
2
$\begingroup$

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

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

$\endgroup$
1
$\begingroup$

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

$\endgroup$
0
$\begingroup$

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

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

$\endgroup$
0
$\begingroup$

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 $$ \frac{3}{\frac{\sqrt{5}}{5}} = 3 \cdot \frac{5}{\sqrt{5}}$$

$\endgroup$
0
$\begingroup$

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 $\dfrac{3}{\frac{\sqrt{5}}{5}}=\dfrac{3}{\frac 1{\sqrt{5}}}=3\sqrt{5}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.