Question

When the following is simplified:
$\left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right)$ is
A. Positive and irrational
B. Positive and rational
C. Negative and irrational
D. Negative and rational

Answer 1

Hint: This question is based on the concept of product of two irrational expressions. To find the product of the two expressions given we will use algebraic formula $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$ where $a = - 2$ and $b = \sqrt 3 $ then simplify it. Then according to the answer obtained we will state the property of the solution from the given options by using definitions of rational and irrationals.

Complete step by step solution:
We have to simplify the expression:
$\left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right)$……$\left( 1 \right)$
Using algebraic formula given below:
$\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$…….$\left( 2 \right)$
On comparing equation (1) and (2) we get,
$a = - 2$, $b = \sqrt 3 $
On substituting the above value in equation (2) we get,
$\Rightarrow \left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right) = {\left( { - 2} \right)^2} - {\left( {\sqrt 3 } \right)^2}$
On simplifying the right side we get,
$\begin{align}
  & \Rightarrow \left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right) = 4 - {\left( 3 \right)^{\dfrac{1}{2} \times 2}} \\
& \Rightarrow \left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right) = 4 - 3 \\
\end{align} $
$\Rightarrow \left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right) = 1$
We got the answer as 1.
So, as 1 is a positive rational number.

So, the correct answer is “Option B”.

Note: We can directly multiply the expression given without using the formula by multiplying each term in one bracket with each term in another bracket and then simplify it as:
$\begin{align}
& \left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right) = - 2 \times - 2 + - 2 \times \sqrt 3 - \sqrt 3 \times - 2 - \sqrt 3 \times \sqrt 3 \\
& \Rightarrow \left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right) = 4 - 2\sqrt 3 + 2\sqrt 3 - 3 \\
& \Rightarrow \left( { - 2 - \sqrt 3 } \right)\left( { - 2 + \sqrt 3 } \right) = 1 \\
\end{align} $
Above method is okay when the expression is not too complicated or the numbers in the expression are not big otherwise it becomes complicated to solve the expression using the above method.
A rational number are those numbers that can be expressed in the form of $\dfrac{p}{q}$ where $q \ne 0$. Decimal expansion of these numbers either terminates after a finite number or the digit starts to repeat them over and over again. If it is not a rational number that means it is an irrational number.