Question

If \[x = 7 + 4\sqrt 3 \], find the value of \[\sqrt x + \dfrac{1}{{\sqrt x }}\].

Answer 1

Hint: Here we have to find the value of the given algebraic expression. For that, we will first find the square root of the given variable. Then we will express the terms under the square root as the square of the sum of two numbers. From there, we will get the value of the square root of the given variable. Then we will find the value of the reciprocal of the square root of the variable. Then we will add both the values to get the required answer.

Complete step-by-step answer:
It is given that:
\[x = 7 + 4\sqrt 3 \]
Taking square root on both sides, we get
\[ \Rightarrow \sqrt x = \sqrt {7 + 4\sqrt 3 } \]
We can write the terms under the square root as
\[ \Rightarrow \sqrt x = \sqrt {7 + 2 \times 2 \times \sqrt 3 } \]
Now, we will write the number 7 as the sum of the number 3 and the number 4. Therefore, we get
 \[ \Rightarrow \sqrt x = \sqrt {4 + 3 + 2 \times 2 \times \sqrt 3 } \]
Now, we will replace the number 4 as \[{2^2}\] and the number 3 as \[{\left( {\sqrt 3 } \right)^2}\].
\[ \Rightarrow \sqrt x = \sqrt {{2^2} + {{\left( {\sqrt 3 } \right)}^2} + 2 \times 2 \times \sqrt 3 } \]
We can see that the terms inside the square root is similar to the algebraic identity \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2 \times a \times b\].
Therefore, using this algebraic identity in the above equation, we get
\[ \Rightarrow \sqrt x = \sqrt {{{\left( {2 + \sqrt 3 } \right)}^2}} \]
On further simplification, we get
\[ \Rightarrow \sqrt x = 2 + \sqrt 3 \] ……… \[\left( 1 \right)\]
Now, we will find the value of \[\dfrac{1}{{\sqrt x }}\].
Now, we will substitute the value of \[\sqrt x \] in \[\dfrac{1}{{\sqrt x }}\]. Therefore, we get
\[ \Rightarrow \dfrac{1}{{\sqrt x }} = \dfrac{1}{{2 + \sqrt 3 }}\]
Now, multiplying the term\[2 - \sqrt 3 \] in numerator and denominator, we get
\[ \Rightarrow \dfrac{1}{{\sqrt x }} = \dfrac{{1 \times \left( {2 - \sqrt 3 } \right)}}{{\left( {2 + \sqrt 3 } \right)\left( {2 - \sqrt 3 } \right)}}\]
Using this algebraic identity \[\left( {a - b} \right)\left( {a + b} \right) = {a^2} - {b^2}\], we get
\[ \Rightarrow \dfrac{1}{{\sqrt x }} = \dfrac{{\left( {2 - \sqrt 3 } \right)}}{{{2^2} - {{\left( {\sqrt 3 } \right)}^2}}}\]
On applying the exponents on the bases, we get
\[ \Rightarrow \dfrac{1}{{\sqrt x }} = \dfrac{{\left( {2 - \sqrt 3 } \right)}}{{4 - 3}}\]
On subtracting the terms in denominator, we get
\[ \Rightarrow \dfrac{1}{{\sqrt x }} = \dfrac{{\left( {2 - \sqrt 3 } \right)}}{1} = 2 - \sqrt 3 \]
We need to calculate the value of \[\sqrt x + \dfrac{1}{{\sqrt x }}\].
Now, we will substitute the value of \[\sqrt x \] and the value of \[\dfrac{1}{{\sqrt x }}\] here.
\[ \Rightarrow \sqrt x + \dfrac{1}{{\sqrt x }} = 2 + \sqrt 3 + 2 - \sqrt 3 \]
On adding and subtracting the numbers, we get
\[ \Rightarrow \sqrt x + \dfrac{1}{{\sqrt x }} = 4\]
Hence, the value of \[\sqrt x + \dfrac{1}{{\sqrt x }}\] is equal to 4.

Note: Here we have used the algebraic identity to find the value of the given algebraic expression. The algebraic equations are defined as the equalities that are valid for all values of variables in them. Algebraic identities are also used for the factorization of polynomials and also algebraic identities are used in the computation of algebraic expressions and solving different polynomials.