simplifying radicals and perfect square numbers?

Question

can any one tell me how i simplfy mixed radicals like 4/5 + 3/15 (pretend the / is a square root symbol) and can some one also give me a list of perfect square numbers.

Answer 1

4sqrt(5) + 3sqrt(15)

This can't be simplified at all; at least, not in the typical way. Let's take a better example:

EXAMPLE: Simplify 4sqrt(2) - 7sqrt(8)

Your first step would be to obtain the prime factorization of the insides of the square roots. The 2 cannot be factorized any further, but the 8 can.

4sqrt(2) - 7sqrt(2 * 2 * 2)

For every TWO occurrences of identical numbers, remove both of the identical occurrances outside of the square root. Upon doing so, the two identical numbers become ONE. In our case, we're going to pull out two 2s, and outside of the square root, it is one 2.

4sqrt(2) - 7[2 sqrt(2)]

Now, we can merge the 2 with the 7, giving us

4sqrt(2) - 14sqrt(2)

And now, we treat the sqrt(2) as if it were a variable x. This gives us

-10sqrt(2).


Back to your example:
4sqrt(5) + 3sqrt(15)

Note that 5 is already prime, so it can't be factorized. 15, however is equal to 3 x 5. As you can see, there are no repeated factors, so there's nothing that can be done with this to simplify it.

Answer 2

4sqrt(5)+3sqrt(3)sqrt(5)

sqrt(5) ( 4 + 3 sqrt(3))

1^2=1
2^2=4
3^2=9
4^2=16
.....