Show clearly that (3√3)^2 = 27

Show clearly that (3√3)² = 27

First of all, the question looks overfacing due to the surds, I can assure you that surds are not scary, they simply act as a means of clearing up messy numbers such as 1.73205 08075 68877 29352 74463... (carries on for a billion digits) which can be simply written as √3! 

So, how would we work out (3√3)² without using a calculator? Well, luckily the question tells us that the answer we need to find is 27, so we can't go far wrong within these parametres. 

First of all, (3√3)²  is a tidy and mathematical way of writing 3√3 x 3√3.

A surd is not an equation, it is just a tidy way of writing a messy number. 

 Remember the rule of surds:

√(a x b) is the same as √a x √b... 

3√3 is a tidy, surdy way of writing √3² x √3. Which is the same as √9 x √3... √(9 x 3).... which can also be written as √27. 

so, what we have to show is that √27²= 27. 

√27 x √27= 27, because, by timesing together two like surds, we can get rid of the root sign because, in general, multiplying two like surds gives a rational number... (√a x √a= a) (2√a x √a=2a) 

Hence, we have just proved that (3√3)²= 27 simply by referring to the basic rules of surds :)