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 n-th root of the result"
Both steps lead back to the a that we started with.
For the simple case where n = 2, thefollowing 4 expressions all have the same value:
√a2 = (√a)2 = √(a2) = a
For example, if a = 9:
√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:
Yet another way of thinking about it is 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
Example:
We could write "the product of the n-th root of a and the n-th root of b is the n-th root of ab" 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
Example:
In words, we would say: "The 4th root of the 3rd root of 5 is equal to the 12th root of 5".
The equivalent general 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)
Example:
If we write the our general expression using fractional exponents, we have:
(b ≠ 0)
Mixed Examples
Simplify the following:
(a) 
(b) 
(c) 
(d) 
Simplest Radical Form Examples
In these examples, we are expressing the answers in simplest radical form, using the laws given above.
(a) 
(b) 
(c) 
(d) 
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.
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