Question

Simplify the following:

`7/(3sqrt(5) - 2) - 2/(3sqrt(5) + 2`

Answer 1

We are asked to simplify the expression:

`7/(3sqrt(5) - 2) - 2/(3sqrt(5) + 2`

Step 1: Rationalise both fractions

We’ll begin by rationalising the denominators of both fractions by multiplying the numerator and denominator of each fraction by the conjugate of the denominator.

First term: `7/(3sqrt(5) - 2)`

Multiply the numerator and denominator by the conjugate of the denominator, `3sqrt(5) + 2`:

`7/(3sqrt(5) - 2) xx (3sqrt(5) + 2)/(3sqrt(5) + 2)`

= `(7(3sqrt(5) + 2))/((3sqrt(5) - 2)(3sqrt(5) + 2))`

Simplify the denominator using the difference of squares formula:

`(3sqrt(5))^2 - 2^2`

= 9 × 5 – 4

= 45 – 4

= 41

Now, expand the numerator:

`7(3sqrt(5) + 2) = 21sqrt(5) + 14`

So the first fraction becomes:

`(21sqrt(5) + 14)/41`

Second term: `2/(3sqrt(5) + 2)`

Multiply the numerator and denominator by the conjugate of the denominator, `3sqrt(5) - 2`:

`2/(3sqrt(5) + 2) xx (3sqrt(5) - 2)/(3sqrt(5) - 2)`

= `(2(3sqrt(5) - 2))/((3sqrt(5) + 2)(3sqrt(5) - 2))`

Simplify the denominator (same as before):

`(3sqrt(5))^2 - 2^2`

= 45 – 4

= 41

Now, expand the numerator:

`2(3sqrt(5) - 2) = 6sqrt(5) - 4`

So the second fraction becomes:

`(6sqrt(5) - 4)/41`

Step 2: Combine the two fractions

Now subtract the two fractions:

`(21sqrt(5) + 14)/41 - (6sqrt(5) - 4)/41`

Since the denominators are the same, we can combine the numerators:

`((21sqrt(5) + 14) - (6sqrt(5) - 4))/41`

Simplify the numerator:

`21sqrt(5) + 14 - 6sqrt(5) + 4`

= `(21sqrt(5) - 6sqrt(5)) + (14 + 4)`

= `15sqrt(5) + 18`

Thus, the expression becomes:

`(15sqrt(5) + 18)/41`