Question

What is the positive root of $7 + 4\sqrt 3 $ ?

Answer 1

Hint: In the given question, we have to evaluate the square root of a number given to us in the problem itself. So, we will use the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ to condense the expression present inside the square root function. Then, we now calculate the root of the square of a number and simplify the expression to get to the required answer. The algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ is used to evaluate the square of a binomial expression involving the sum of two terms.

Complete step by step answer:
Given question requires us to find the value of the positive square root of $7 + 4\sqrt 3 $.
So, we will split up the entity into separate parts to resemble the square of a binomial. We will use the algebraic identity ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$ to calculate the positive root of $7 + 4\sqrt 3 $.
So, we have, \[7 + 4\sqrt 3 = 7 + 2\left( 2 \right)\left( {\sqrt 3 } \right)\]
Splitting up $7$ as $3 + 4$,
\[ \Rightarrow 7 + 4\sqrt 3 = 4 + 3 + 2\left( 2 \right)\left( {\sqrt 3 } \right)\]
\[ \Rightarrow 7 + 4\sqrt 3 = {\left( 2 \right)^2} + {\left( {\sqrt 3 } \right)^2} + 2\left( 2 \right)\left( {\sqrt 3 } \right)\]
Now, we use the algebraic identity for the square of sum of two terms ${\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}$.
\[ \Rightarrow 7 + 4\sqrt 3 = {\left( {2 + \sqrt 3 } \right)^2}\]
Now, we have to find the square root of this whole square expression.
So, we get, \[\sqrt {{{\left( {2 + \sqrt 3 } \right)}^2}} \]
Now, the square root function cancels the square. We know that the square root value is always positive. So, we get,
\[ \Rightarrow \left( {2 + \sqrt 3 } \right)\]
Positive square root of $7 + 4\sqrt 3 $ is \[\left( {2 + \sqrt 3 } \right)\].

Note:
Before attempting such questions, one should memorize all the algebraic identities and should know their applications in such problems. Care should be taken while carrying out the calculations. We can also verify the answer of the given question by doing the solution backwards and calculating the square of \[\left( {2 + \sqrt 3 } \right)\].