Question

Simplify the following to the simplest form: $\left| \sqrt{7-4\sqrt{3}} \right|$

Answer 1

Hint: Try to convert $7-4\sqrt{3}$ to a perfect square to easily remove the square root and simplify. The simplification requires the knowledge of rational and irrational numbers along with the square root and modulus operators.

Complete step-by-step answer:
For our convenience, we let $7-4\sqrt{3}$ to be ${{x}^{2}}$.
So, our questions becomes:
$\left| \sqrt{7-4\sqrt{3}} \right|=\left| \sqrt{{{x}^{2}}} \right|................(i)$
To proceed in the question, try to simplify ${{x}^{2}}$ ;
${{x}^{2}}=7-4\sqrt{3}$
Breaking 7 as sum of 3 and 4.
We have;
${{x}^{2}}=3+4-4\sqrt{3}$
Here, the terms can be written as:
$3=\sqrt{3}.\sqrt{3}={{\left( \sqrt{3} \right)}^{2}}$
$4={{2}^{2}}$
$4\sqrt{3}=2.2\sqrt{3}$
So, our equation becomes:
${{x}^{2}}=3+4-4\sqrt{3}$
$\Rightarrow {{x}^{2}}={{\sqrt{3}}^{2}}+{{2}^{2}}-2.2\sqrt{3}$
Now, we know:
${{\left( a-b \right)}^{2}}={{b}^{2}}+{{a}^{2}}-2ab$
Using the above formula, we get:
${{x}^{2}}={{\sqrt{3}}^{2}}+{{2}^{2}}-2.2\sqrt{3}$
$\Rightarrow {{x}^{2}}={{\left( 2-\sqrt{3} \right)}^{2}}$
Now, moving back to equation (i);
$\left| \sqrt{7-4\sqrt{3}} \right|=\left| \sqrt{{{x}^{2}}} \right|$
$\Rightarrow \left| \sqrt{7-4\sqrt{3}} \right|=\left| \sqrt{{{\left( 2-\sqrt{3} \right)}^{2}}} \right|$
We know;
$\left| \sqrt{{{k}^{2}}} \right|=\left| \pm k \right|$
Applying we get;
$\left| \sqrt{7-4\sqrt{3}} \right|=\left| \pm \left( 2-\sqrt{3} \right) \right|$
And we know, “the result of subtraction of two real numbers is a real number.”
Therefore, we can state that $2-\sqrt{3}$ is real.
We also know the approximate value of $\sqrt{3}$ is 1.7 so we can say that $2-\sqrt{3}$ is positive as well.
And we know, “the absolute value of any real number is a positive real number.”
So, combining this statements and the equation we get,
$\left| \sqrt{7-4\sqrt{3}} \right|=\left| \pm \left( 2-\sqrt{3} \right) \right|$
$\Rightarrow \left| \sqrt{7-4\sqrt{3}} \right|=\left( 2-\sqrt{3} \right)$
Hence, the simplified form of $\left| \sqrt{7-4\sqrt{3}} \right|$ is $2-\sqrt{3}$.

Note: At the first glance of the question you might think that you can directly put in the value of$\sqrt{3}$ and get the answer but that won’t be simplification nor you will be able to get an accurate answer as the answer you would get is approximate of the exact one because you cannot put the exact value of $\sqrt{3}$ as it’s a non-terminating decimal number i.e. have infinite numbers occurring after the decimal points.
In questions including roots you can also give a try to rationalisation to get a simplified result or to initiate the simplification process.