3. Simplest Radical Form
Before we can simplify radicals, we need to know some rules about them. These rules just follow on from what we learned in the first 2 sections in this chapter, Integral Exponents and Fractional Exponents.
Expressing in simplest radical form just means simplifying a radical so that there are no more square roots, cube roots, 4th roots, etc left to find. It also means removing any radicals in the denominator of a fraction.
Laws of Radicals
n-th root of a Number to the Power n
We met this idea in the last section, Fractional Exponents. Basically, finding the n-th root of a number is the opposite of raising the number to the power n, so they effectively cancel each other out. These 4 expressions have the same value:
The 2nd item in the equality above means:
"take the nth root first, then raise the result to the power n"
The 3rd item means:
"raise a to the power n then find the nth root of the result"
Both steps lead back to the a that we started with.
For the simple case (where n = 2), these all have the same value:
√a2 = (√a)2 = √(a2) = a
For example:
√92 = (√9)2 = √(92) = 9
The second item means find the square root of 9 (answer: 3) then square it (answer 9).
The 3rd item means square 9 first (we get 81) then find the square root of the result (answer 9).
We could write all this using fractional exponents as follows:
The Product of the n-th root of a and the n-th root of b is the n-th root of ab
We could write this using fractional exponents as well:
The m-th Root of the n-th Root of the Number a is the mn-th Root of a
The equivalent expression using fractional exponents is as follows:
The n-th Root of a Over the n-th Root of b is the n-th Root of a/b
(b ≠ 0)
If we write the same thing using fractional exponents, we have:
(b ≠ 0)
Example 1.
Simplify the following:
(a) 
Answer:
We have used the first law above.
(b) 
Answer:
We have used 
.
(c) 
Answer:
We have used the law: 
(d) 
Answer:
Nothing much to do here. We used:
Example 2
In these examples, we are expressing the answers in simplest radical form, using the laws given above.
(a) 
Answer:
We need to examine 72 and find the highest square number that divides into 72. (Squares are the numbers 12 = 1, 22 = 4, 32 = 9, 42 = 16, ...)
In this case, 36 is the highest square that divides into 72 evenly. We express 72 as 36 × 2 and proceed as follows.
We have used the law:
(b) 
Answer:
We have used the law: √a2 = a
(c) 
Answer:
(d) 
Answer:
Exercises. Simplify:
Q1 
Q2 
Q3 
This one requires a special trick. To remove the radical in the denominator, we need to multiply top and bottom of the fraction by the denominator.
Book mark this page in Del.icio.us, Furl, Digg, StumbleUpon, whatever...
Didn't find what you are looking for? Try search:
Need a break? Play a math game. Well, they all involve math... No, really!
Home |
Sitemap |
About & Contact |
Feedback & questions |
Privacy
Users online:
30. Highest ever: 68, on 20 May 2008
Page last modified: 14 May 2008
Valid HTML 4.01
| Valid CSS