Question

Examine whether the following numbers are rational or irrational
(3+√2)(2-√3)(3-√2)(2+√3)​

Answer 1

Step-by-step explanation:

(3+√2)(2-√3)(3-√2)(2+√3)

= (3+√2)(3-√2)(2+√3)(2-√3)

= [(3)²-(√2)²] [(2)²-(√3)²]

= [9-2] [4-3]

= 7×1 = 7.

Since, 7 is rational.

So, (3+√2)(2-√3)(3-√2)(2+√3) is also rational.

Answer 2

The following numbers are rational or irrational (3+√2)(2-√3)(3-√2)(2+√3)​, i.e. equal to 7, is a Rational Number.

We are going to use the Following to Major Concepts to Solve the Problem:

1) The Associative Law of Multiplication and Addition

• The parentheses in the Associative Law move but the numbers or characters do not.
• The Associative Law works only when we Multiply or Add. But When we divide or subtract it DOES NOT function.
• [a x b] x [c x d] = [a x c] x [b x d]  = [a x d] x [b x c]

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

Applying The Associative Law of Multiplication to the question, we get

(3+√2)(2-√3)(3-√2)(2+√3)​

= (3+√2)(3-√2) x (2-√3)(2+√3)​

Now, using the Formulae:

= (3+√2)(3-√2)(2+√3)(2-√3)

= [(3)²-(√2)²] x [(2)²-(√3)²]

= [9-2] x [4-3]

= 7×1

= 7

Rational Number:

Any number that can be expressed as a fraction, i.e. in the form of $\frac{x}{y}$, with $'\ y\ '$ being the denominator and $'\ x\ '$ being the numerator, is said to be a Rational Number if $y\neq 0$.

Therefore,

The following numbers are rational or irrational (3+√2)(2-√3)(3-√2)(2+√3)​, i.e. equal to 7, is a Rational Number.

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