Question

If \[\sqrt {13 - x\sqrt {10} } = \sqrt 8 + \sqrt 5 \], then what is the value of \[x\]?
A) \[ - 5\]
B) \[ - 6\]
C) \[ - 4\]
D) \[ - 2\]

Answer 1

Hint:
Here, we will square both the sides of the given equation to remove the square root on the left hand side of the equation. Then we will use the appropriate algebraic identity to solve the right hand side of the equation. We will solve the equation further to get the required value of \[x\].

Complete Step by step Solution:
First, we will remove the square root from the left-hand side so that we can get our\[x\] with integer power.
The given equation is \[\sqrt {13 - x\sqrt {10} } = \sqrt 8 + \sqrt 5 \].
As according to the formula \[\sqrt[n]{x} = {x^{\dfrac{1}{n}}}\] , we have \[n = 2\] on the left side of the equation.
So to make it a rational power we will square both sides as follows:
\[\begin{array}{l} \Rightarrow {\left( {\sqrt {13 - x\sqrt {10} } } \right)^2} = {\left( {\sqrt 8 + \sqrt 5 } \right)^2}\\ \Rightarrow {\left( {13 - x\sqrt {10} } \right)^{\dfrac{1}{2} \times 2}} = {\left( {\sqrt 8 + \sqrt 5 } \right)^2}\end{array}\]\[\]
Using the formula \[{\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab\] on right side, we get
\[ \Rightarrow 13 - x\sqrt {10} = {\left( {\sqrt 8 } \right)^2} + {\left( {\sqrt 5 } \right)^2} + 2\sqrt 8 \sqrt 5 \]
Applying the exponent on the terms, we get
\[ \Rightarrow 13 - x\sqrt {10} = 8 + 5 + 2\sqrt {40} \]
Adding the like terms, we get
\[ \Rightarrow 13 - x\sqrt {10} = 13 + 2\sqrt {40} \]
Subtracting 13 from both sides, we get
\[\begin{array}{l} \Rightarrow - x\sqrt {10} = 2\sqrt {40} \\\end{array}\]
We can write \[\sqrt {40} \] as\[\sqrt {4 \times 10} = 2\sqrt {10} \]. So, we get
\[ \Rightarrow x = - \dfrac{{2 \times 2\sqrt {10} }}{{\sqrt {10} }}\]
Simplifying the expression, we get
\[ \Rightarrow x = - 4\]

Hence, option (c) is correct.

Note:
In order to solve this question, we try to remove the fraction power or square root first from the variables. The rational exponent in \[\sqrt[n]{x} = {x^{\dfrac{1}{n}}}\] is \[\dfrac{1}{n}\] where 1 is defined as the power and \[n\] is defined as the root index It is necessary to remove square root from the term having variable, in this case, \[x\] so that our equation doesn’t get too complicated. The equation that has formed after removing the square root is a linear equation. A linear equation is an equation, which has the highest degree of variable as 2 and has only one solution.