5 square root of 3 times 8 square root of 21. multiplying/dividing/ simplifying radicals

Question

really need help answering this problem

Answer 1

So we have 5√3 x 8√21... The easiest thing to do with a problem like this is to look at the factors for the numbers in the radicals first.

 

3 doesn't give you a lot as it is prime so it is just 3 and 1. 

 

21 is interesting in that its factors are 3 and 7 both are primes though so we have to stop here. 

 

Normally the overall strategy is to try to find factors that are squares (e.g., 4, 9, 16, etc.). But here the fact that 3 is a factor is useful given the numbers we are trying to simplify. 

 

First we break 21 up into its factors (keeping them under the radical) 

 

5√3 x 8√21 becomes.... (5)(√3)(8)(√3)(√7) [just changed the notation slightly, but the same meaning is there and you can see 21 has been broken down into its two factors] 

 

Now (√3)(√3) or (√3)² = 3. From here it should be easier to simplify... 

 

(5)(3)(8)(√7) = 120(√7)