Question

state whether the number (√2-√3)(√2+√3) is rational or irrational justify

Answer 1

Given: we have given a number $(\sqrt2-\sqrt3)(\sqrt2+\sqrt3)$.

To find: we have to justify it is rational or irrational

Step by step solution:

The rational numbers are express in the term of p/q

where p and q are integers.

Now, we have the given number is$(\sqrt2-\sqrt3)(\sqrt2+\sqrt3).$

Using the identity, $(a-b)(a+b) = a^2-b^2$

$(\sqrt2-\sqrt3)(\sqrt2+\sqrt3) = (\sqrt2^2-\sqrt3^2) \\= (2-3)\\ = -1$

So, the number -1 can be expressed in the form of p/q i.e -1/1.

Final answer:

Hence the given number is the rational.

Answer 2

Rational number, in arithmetic, a number that can be represented as the quotient p/q of two integers such that q ≠ 0. In addition to all the fractions, the set of rational numbers includes all the integers, each of which can be written as a quotient with the integer as the numerator and 1 as the denominator.

An irrational number is a type of real number which cannot be represented as a simple fraction. It cannot be expressed in the form of a ratio. If N is irrational, then N is not equal to p/q where p and q are integers and q is not equal to 0. Example: √2, √3, √5, √11, √21, π(Pi) are all irrational.

to solve the given problem we will apply the identity $(a-b)(a+b)=a^{2}-b^{2}$ and then on putting the given values using above identify, we will check whether we get rational or irrational.

So, $(\sqrt{2}-\sqrt{3}) (\sqrt{2}+\sqrt{3})$

$(a+b)(a-b)=a^{2}-b^{2}$

$=(\sqrt{2})^2-(\sqrt{3})^2$

$=2-3=-1$

So, we get $(\sqrt{2}-\sqrt{3}) (\sqrt{2}+\sqrt{3})=-1$

We know that $(-1)$ is a negative integer and can be expressed as $-1$, hence it is a rational number $1$.

Thus, the number $(\sqrt{2}-\sqrt{3}) (\sqrt{2}+\sqrt{3})$ is a rational number.