Final answer:
To express √20 * √8 in simplest radical form, multiply the numbers inside the radicals to get √160. After simplifying the square root of 160, which is 4√10, we find that the final simplest form is 4√5.
Step-by-step explanation:
The question asks us to express the product of two square roots, √20 and √8, in simplest radical form. To do this we must first multiply the numbers under the square roots:
√20 * √8 = √(20 * 8) = √160
Now, let's find the prime factorization of 160 to simplify the radical:
160 = 2 * 80
160 = 2 * (2 * 40)
160 = 2 * (2 * (2 * 20))
160 = 2 * (2 * (2 * (2 * 10)))
160 = 2 * (2 * (2 * (2 * (2 * 5))))
160 = 2^5 * 5
Since we are dealing with square roots, we look for pairs of prime factors. We have a pair of 2's, actually, two pairs, which we can take out of the square root:
√160 = √(2^5 * 5) = √(2^4 * 2 * 5) = 2^2 * √(2 * 5) = 4√10
However, we're not yet finished because 10 itself has a square factor:
10 = 2 * 5
√10 = √(2 * 5)
Thus we can simplify further:
4√10 = 4 (√2 * √5) = 4√2√5
Since there are no square factors left, this is our final simplified form. But let's revert to the original request for easier comparison:
4√2√5 = (4√2) √5 = 2√2 * 2√5 = (2*2)√5 = 4√5