Question

State whether true or false in the following: 

`1/(3 - sqrt(8)) = 3 + 2sqrt(2)`

Answer 1

This statement is true.

Explanation:

Step 1: Simplify the denominator

Note that `sqrt(8) = 2sqrt(2)`

So, `1/(3 - sqrt(8)) = 1/(3 - 2sqrt(2))`

Step 2: Rationalise the denominator

Multiply numerator and denominator by the conjugate `3 + 2sqrt(2)`:

`1/(3 - 2sqrt(2)) xx (3 + 2sqrt(2))/(3 + 2sqrt(2))`

= `(3 + 2sqrt(2))/((3 - 2sqrt(2))(3 + 2sqrt(2))`

Step 3: Simplify denominator

`(3 - 2sqrt(2))(3 + 2sqrt(2))`

= `3^2 - (2sqrt(2))^2`

= 9 – 8

= 1

Step 4: Final simplification

`(3 + 2sqrt(2))/1 = 3 + 2sqrt(2)`

Step 5: Compare both sides

LHS = `1/(3 - sqrt(8))`

RHS = `3 + 2sqrt(2)`

They are equal.