Simplify each of the following expressions:
(i) (3 + √3)(2 + √2)
(ii) (3 + √3)(3 - √3)
(iii) (√5 + √2 )2
(iv) (√5 - √2)(√5 + √2)
(i) \( (3 + \sqrt{3})(2 + \sqrt{2}) \):
Using the distributive property: \[ (3 + \sqrt{3})(2 + \sqrt{2}) = 3(2) + 3(\sqrt{2}) + \sqrt{3}(2) + \sqrt{3}(\sqrt{2}) \] Simplifying each term: \[ = 6 + 3\sqrt{2} + 2\sqrt{3} + \sqrt{6} \] So, the simplified expression is: \[ 6 + 3\sqrt{2} + 2\sqrt{3} + \sqrt{6} \]
(ii) \( (3 + \sqrt{3})(3 - \sqrt{3}) \):
This is a difference of squares: \[ (a + b)(a - b) = a^2 - b^2 \] Here, \( a = 3 \) and \( b = \sqrt{3} \), so: \[ (3 + \sqrt{3})(3 - \sqrt{3}) = 3^2 - (\sqrt{3})^2 \] Simplifying: \[ = 9 - 3 = 6 \] So, the simplified expression is: \[ 6 \]
(iii) \( (\sqrt{5} + \sqrt{2})^2 \):
Using the expansion \( (a + b)^2 = a^2 + 2ab + b^2 \): \[ (\sqrt{5} + \sqrt{2})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{2}) + (\sqrt{2})^2 \] Simplifying each term: \[ = 5 + 2\sqrt{10} + 2 \] So, the simplified expression is: \[ 7 + 2\sqrt{10} \]
(iv) \( (\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) \):
This is again a difference of squares: \[ (a - b)(a + b) = a^2 - b^2 \] Here, \( a = \sqrt{5} \) and \( b = \sqrt{2} \), so: \[ (\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2}) = (\sqrt{5})^2 - (\sqrt{2})^2 \] Simplifying: \[ = 5 - 2 = 3 \] So, the simplified expression is: \[ 3 \]
The simplified expressions are:
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 _____.