Question

Find the positive square root of \[7+\sqrt{48}\] is:

Answer 1

Hint: Now here in this question to solve it we will start by taking the root of the expression given to us in the question. Now we will use the logic of factoring to see if we can convert it to a form whose root is findable to us. Now here in this question we try to convert it into the form of a square in addition to two numbers of formulas because that’s the format the question is. Therefore here to find the root we will try to convert the question in the form of
\[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\]

Complete step-by-step answer:
Now here in this question the expression given to us is
\[7+\sqrt{48}\]
Now it is asked of us to find the root of this expression therefore we can express it as
\[\Rightarrow \sqrt{7+\sqrt{48}}\]
Now since \[48\] can be factorized therefore to solve this further and write this in a different way we can write it as
\[\Rightarrow \sqrt{7+\sqrt{16\times 3}}\]
Now we know the root of \[16\] is \[4\] therefore taking it out of the root we get
\[\Rightarrow \sqrt{7+4\times \sqrt{3}}\]
Now here we also know that we can express \[7\] to be the addition of two numbers. Now since we need to express \[7\] as the square of \[\sqrt{3}\] so as to make it possible to solve it further so we can write \[7\] here in this question as
\[\Rightarrow \sqrt{3+4+4\times \sqrt{3}}\]
Now we know that \[4\] is the square of \[2\] hence we can express it as
\[\Rightarrow \sqrt{3+{{2}^{2}}+2\times 2\times \sqrt{3}}\]
Now we can also right \[3\] here in the form of square of its root which is
\[\Rightarrow \sqrt{{{\left( \sqrt{3} \right)}^{2}}+{{\left( 2 \right)}^{2}}+2\times 2\times \sqrt{3}}\]
Now as we can see that the expression under the root here is in the form of \[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\] with a and b being \[2\] and \[\sqrt{3}\] respectively there we can now write the expression as
\[\Rightarrow \sqrt{{{\left( 2+\sqrt{3} \right)}^{2}}}\]
Now as we know that the root of a square is the number itself therefore the positive root of the question is \[2+\sqrt{3}\].
 \[\Rightarrow \sqrt{7+\sqrt{48}}=2+\sqrt{3}\]
So, the correct answer is “\[2+\sqrt{3}\]”.

Note: Now a tip to solve this question is that to simplify a square root we need to make the number inside it as small as possible by factoring it and finding its HCF. The positive number when multiplied by itself gives us the square of a number. Taking the square root of a square of a positive number gives us the original number itself.