Question

State with reason, whether the following is a surd and which is not?

`2sqrt(3) xx 6sqrt(27)`

Answer 1

Not Surd

Reason:

Given expression: `2sqrt(3) xx 6sqrt(27)`

Step-wise calculation:

1. Simplify the surds inside the square roots:

`sqrt(27)`

= `sqrt(9 xx 3)`

= `3sqrt(3)`

So, `6sqrt(27)`

= `6 xx 3sqrt(3)`

= `18sqrt(3)`

2. Now multiply the terms: 

`2sqrt(3) xx 18sqrt(3)`

= `(2 xx 18)(sqrt(3) xx sqrt(3))`

= 36 × 3

= 108

Since the product simplifies to a rational number 108, neither `2sqrt(3)` nor `6sqrt(27)` individually becomes rational, but the original expression simplifies to a rational number.

`2sqrt(3)` is a surd because `sqrt(3)` is irrational.

`6sqrt(27)` is a surd since `sqrt(27)` is irrational.

Their product `2sqrt(3) xx 6sqrt(27)` is a rational number 108, so the expression itself is not a surd.

Thus, the given expression simplifies into a rational number and is not a surd, though the individual terms are surds.