Question

Is the number $\left( {3 + \sqrt 7 } \right)\left( {3 - \sqrt 7 } \right)$ rational or irrational.

Answer 1

Hint:
At first we need to simplify the number$\left( {3 + \sqrt 7 } \right)\left( {3 - \sqrt 7 } \right)$ using the identity $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$ and we need to check whether the number can be written in the form of $\dfrac{p}{q}$. If so it is rational, otherwise it is irrational.

Complete step by step solution:
We are given a number $\left( {3 + \sqrt 7 } \right)\left( {3 - \sqrt 7 } \right)$
Now first let's simplify the given number
We have the identity $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$
Here a = 3 and b = $\sqrt 7 $
Hence by using the property we get,
$
   \Rightarrow \left( {3 + \sqrt 7 } \right)\left( {3 - \sqrt 7 } \right) = {\left( 3 \right)^2} - {\left( {\sqrt 7 } \right)^2} \\
   \Rightarrow \left( {3 + \sqrt 7 } \right)\left( {3 - \sqrt 7 } \right) = 9 - 7 = 2 \\
$
Hence we get the value of the expression to be 2.
The number 2 can be written in the form of $\dfrac{p}{q}$.
That is $\dfrac{2}{1}$.

Hence the given number is a rational number.

Note:
1) Many students tend to say that it is irrational as it contains an irrational $\sqrt 7$.
2) But only after simplifying we get that it is not an irrational number.
3) Irrational numbers cannot be expressed in fractions. Rational numbers include perfect squares.
4) Rational numbers include decimals which are finite and repeating.