Question

The number \[\left( {3 + \sqrt 3 } \right)\left( {3 - \sqrt 3 } \right)\] is a/an ......…
A. Rational number
B. Natural number
C. Real number
D. All of these

Answer 1

Hint: In this given question, we have been given an expression which involves a natural number and an irrational number. On the first sight it seems that the answer is going to be an irrational number. But the other bracket also contains a similar set of numbers and we do not just jump on the conclusion that the value of the whole expression is an irrational answer. First, we need to check if the system of expression represents an identity with which we can solve that expression with ease. Then we apply the formula of the identity to the expression and evaluate it under the bounds and calculate the answer and then check under which bracket (rational, irrational, natural, et cetera) the answer lies.

Formula Used:
We are going to apply the formula of the difference of squares on the expression, which is:
\[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\]

Complete step-by-step answer:
We have to evaluate the value of \[\left( {3 + \sqrt 3 } \right)\left( {3 - \sqrt 3 } \right)\].
Now, this expression is similar to the formula of the identity of difference of squares, which is:
\[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\]
So, we put in the values of the expression as \[a = 3\] and \[b = \sqrt 3 \] into the identity and we get:
\[\left( {3 + \sqrt 3 } \right)\left( {3 - \sqrt 3 } \right) = {3^2} - {\left( {\sqrt 3 } \right)^2} = 9 - 3 = 6\]
Hence, the answer of \[\left( {3 + \sqrt 3 } \right)\left( {3 - \sqrt 3 } \right)\] is 6.
Now, this answer is clearly a natural number. But we know that any natural number is a rational number and also that, any rational number is a real number. Hence, any natural number is a rational as well as a real number. Hence, our answer is a natural, rational and a real number.

Hence, the correct option is D).

Note: So, we saw that in solving questions like these, we first write down what has been given to us. Then we check if the expression given in the question is relatable to any identity which might help us solve the given expression. Then we put in the values of the given expression into the identity and then we finally solve the expression and arrive at the answer.