Question

If a = 3 + 2√2, then find the value of a 2 + 1/a 2

Answer 1

Step-by-step explanation:

Given:-

a = 3+2√2

To find:-

Find the value of a^2+(1/a^2)?

Solution:-

Given that

a = 3+2√2

=> 1 / a = 1 / (3+2√2)

The denominator = 3+2√2

We know that

The Rationalising factor of a+√b = a-√b

The Rationalising factor of 3+2√2 = 3-2√2

1/a = [(1/3+2√2)]×[(3-2√2)/(3-2√2)]

=> 1/a = (3-2√2)/(3+2√2)(3-2√2)

=> 1/a = (3-2√2)/[3^2-(2√2)^2]

=> 1/a = (3-2√2)/[9-8]

=> 1/a = 3-2√2/1

=> 1/a = 3-2√2

Now

a+ (1/a)

=> 3+2√2+3-2√2

=> (3+3)+(2√2-2√2)

=> 6+0

=> 6

a+(1/a) = 6

On squaring both sides then

[a+(1/a)]^2 = 6^2

=> a^2+(1/a)^2+2(a)(1/a) = 36

=> a^2+(1/a^2)+2 = 36

=> a^2+(1/a^2) = 36-2

a^2+(1/a^2) = 34

Answer:-

The value of a^2+(1/a^2) for the given problem is 34

Used formulae:-

• The Rationalising factor of a+√b = a-√b

• (a+b)(a-b)=a^2-b^2

• (a+b)^2=a^2+2ab+b^2