Ask Me Help Desk - Simplifying numbers completely

How would I completely simplify these?

a) √10 x √8 I know that it is like 80x or maybe even 40x..

b) √24x^4 (24 x to the 4th power) / √3x I think it might be like 2x√2x?

c) √20 + 6√5 I'm guessing it might be 6√5 or 12√5?

When multiplying square roots of real numbers like these you simply multiply all the things that are under a square root sign and put the result under a square root, and then multiply those things that are not under a square root as normal. In other words:

$ a \sqrt b \ \times \ c \sqrt d = ab \sqrt {cd} $

I'll help you with an example that is similar to problem (c): consider

$ 3 \sqrt {32} \times 5 \sqrt 2 = 3 \time 5 sqrt {32 \times 2} = 15 \sqrt {64} = 15 \times 8 = 120. $

Hope this helps.

For b, if this is the problem:

$ \frac {\sqrt {24 x^4}} {\sqrt {3x}} $

then you can proceed like this:

$ \frac {\sqrt {24 x^4}} {\sqrt {3x}} = \sqrt {\frac {24} 3 \frac {x^4} x } = \sqrt {8 x^3} = \sqrt {4x^2 \times 2 x} = 2x \sqrt {2x} $