Verified answer

[tex] \frac{3 + \sqrt{2} }{3 - \sqrt{2} } =a + b \sqrt{2} \\ \frac{3 + \sqrt{2} }{3 - \sqrt{2} } \times \frac{3 + \sqrt{2} }{3 + \sqrt{2} } =a + b \sqrt{2}\\ \frac{(3 + \sqrt{2})(3 + \sqrt{2} ) }{(3 - \sqrt{2} )(3 + \sqrt{2}) } =a + b \sqrt{2} \\ \frac{ {(3 + \sqrt{2} )}^{2} }{ {(3)}^{2} - {( \sqrt{2} )}^{2} } =a + b \sqrt{2} \\ \frac{ {(3)}^{2} + {( \sqrt{2} )}^{2} + 2(3)( \sqrt{2} ) }{ {(3)}^{2} - {( \sqrt{2} )}^{2} } =a + b \sqrt{2}\\ \frac{9 + 2 + 6 \sqrt{2} }{9 - 2} =a + b \sqrt{2} \\ \frac{11 + 6 \sqrt{2} }{7} =a + b \sqrt{2} \\ \frac{11}{7} + \frac{6 \sqrt{2} }{7} =a + b \sqrt{2} \\ a = \frac{11}{7} \: and \: b = \frac{6}{7} [/tex]

Answer:

a= 11/7

b = 6/7

Step-by-step explanation:

3+√2/3-√2 = a+b√2

Step 1- Rationalize The Denominator

Conjugate of 3-√2 = 3+√2

3+√2 x  3+√2

3-√2   x  3+√2

= ( 3+√2)²      ...... Using (a+b)² = a²+b²+2ab

  (3)² - (√2)²   .... Using Identity (a+b)(a-b) = a²-b²

=  9 +2 + 6√2

   9-2

= 11+6√2

      7

a= 11/7

b = 6/7