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Question

Simplify by combining like terms:
5sqrt8 + 2sqrt18

Answer 1

Both square roots can be simplified.

8 = 4 * 2, so sqrt(8) = sqrt(4 * 2)

sqrt(a*b) = sqrt(a)*sqrt(b), so sqrt(4 * 2) = sqrt(4)*sqrt(2)

sqrt(4) = 2, so sqrt(4)*sqrt(2) = sqrt(2)

5sqrt(8) = 10sqrt(2).

Similarly:

sqrt(18) = sqrt(9 * 2) = sqrt(9)*sqrt(2) = 3sqrt(2)

So 2sqrt(18) = 6sqrt(2)

5sqrt(8) + 2sqrt(18) = 10sqrt(2) + 6sqrt(2) = 16sqrt(2)

Because we had terms multiplied by the square root of 2, we could add them together. Basically, we "counted" the number of "square roots of 2" contained in all of the numbers.

Answer 2

Note that

5sqrt(8) + 2sqrt(18)
5 sqrt(2*4) + 2 sqrt (2*9)
5 sqrt(2)sqrt(4) + 2 sqrt(2)sqrt(9)
5 sqrt(2) (2) + 2 sqrt(2) (3)
10 sqrt(2) + 6 sqrt(2) = 16 sqrt(2)

Answer 3

see if there are any perfect squares in 8 and 18
8=2*2*2
18=2*3*3
there are, 4 & 9
find the square roots of those numbers
2 & 3, respectively
multiply outside number (number not in sqrt thing, or radical) by their respective numbers and "remove" them from the square roots to get
10sqrt2 + 6sqrt2
since they are both like terms now, you can add them to get 16sqrt2

Answer 4

sqrt(ab) = sqrt(a) * sqrt(b)

so ... 5sqrt8 + 2sqrt18 =
5(sqrt(2*2*2) + sqrt(4)(sqrt(2*3*3*)) =
5(sqrt(2*2*2) + (sqrt(2*2*2*3*3*) =
5(sqrt(2*2*2) + 3(sqrt(2*2*2)) =
(5 + 3)(sqrt(2*2*2) =
8(sqrt(4*2) =
16(sqrt(2))

Make sense?