Simplifying square roots and square roots in fractions.

The editing software for the blog does not make it easy to write a square root sign (also known as a radical), so instead I will write sqrt(n) to denote the square root of n.

Square roots of perfect squares

Since 5x5 = 25, which we can also write as 5² = 25, we can simplify sqrt(25) = 5. Any perfect square inside a square root is written more simply as the original number before it was squared then the root was taken, assuming the original number was not negative.

Simplifying the square roots of numbers that are not perfect squares
Consider sqrt(24), which is not a perfect square.  We can find the prime factorization of 24 by a factor tree, a method shown in class, and get that 24 = 2×2×2×3.  If we take sqrt(2×2×2×3), we have a square, 2×2, under the root sign, so the pair may be removed and one is put outside the root sign.  (Here I used the metaphor taught to me so long ago by mean Mrs. Kruger, that the root sign is like a prison, and to escape you need a twin.  One twin escapes and the other is killed, never to be seen again.)  Using the prison break rule.  sqrt(2×2×2×3) = 2×sqrt(2×3) = 2sqrt(6).

Simplifying fractions with square roots
To put a fraction involving a square root in lowest terms, the denominator must be free of radical signs.  For instance, 1/sqrt(n) is not in correct form, so we multiply top and bottom by sqrt(n) and get sqrt(n)/n.

Example: 1/sqrt(8) can be changed to sqrt(8)/8, but 8 = 2×2×2, so sqrt(8) is 2sqrt(2).  Since the 2 in front of the radical is not "in the prison", we can reduce the fraction 2sqrt(2)/8 = sqrt(2)/4.

Practice problems

a) sqrt(50)
b) sqrt(48)
c) sqrt(99)
d) sqrt(98)
e) 1/sqrt(48)
f) sqrt(98)/sqrt(50)

Answers in the comments.