Classify the following numbers as rational or irrational:
(i) \(2 - \sqrt5\)
(ii) \((3 + \sqrt23) - \sqrt23\)
(iii) \(\frac{2 \sqrt{7}} { 7 \sqrt7}\)
(iv) \(\frac{1}{\sqrt{2}}\)
(v) 2π
(i) \( \frac{2}{5} \): This is a rational number because it can be expressed as a fraction of two integers.
(ii) \( (3 + \sqrt{3}) - \sqrt{3} \): Simplifying: \[ (3 + \sqrt{3}) - \sqrt{3} = 3 \] Since 3 is an integer, this is a rational number.
(iii) \( \sqrt{7} \): This is an irrational number because the square root of 7 is not a perfect square and cannot be expressed as a fraction of two integers.
(iv) \( \frac{1}{2} \): This is a rational number because it is a fraction of two integers.
(v) \( 2\pi \): This is an irrational number because \( \pi \) is irrational, and multiplying an irrational number by a rational number (2) still results in an irrational number.
The classification of the numbers is:
For real number a, b (a > b > 0), let
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \leq a^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \geq 1 \right\} = 30\pi\)
and
\(\text{{Area}} \left\{ (x, y) : x^2 + y^2 \geq b^2 \text{{ and }} \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 \right\} = 18\pi\)
Then the value of (a – b)2 is equal to _____.