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Radicals and Roots
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Nth roots, fractional and rational exponents, conjugates.
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nth Roots
1. Prior learning
For any number a and positive integer n,
n√a
is called the nth root of a.
The nth root of a is the number that gives you a when raised to the nth power:
(n√a)n=a
If n is even, then (n√a)n is necessarily positive, so we must restrict a>0.
Roots of negative numbers
SL Core 1.5
If a is negative, n√a is negative for all odd n.
For even n, no real n√a exists.
Converting nth roots to fractional exponents
SL AA 1.7
Roots can always be written as fractional exponents and vice versa:
n√a=an1.
Rational exponents
SL AA 1.7
Utilizing nth roots and exponential laws we can rewrite any rational exponent:
anm=(an1)m=n√am=(n√a)m
Simplest form radicals
SL AA 1.7
A radical is in simplest form if the integer under the radical sign is as small as possible.
For example, the simplest form of √48 is 4√3. We can simplify by splitting the radical into a reducible and irreducible part:
√48=√16⋅√3=4√3.
Simplest form fractions with radicals (multiplying by roots)
SL AA 1.7
A fraction in simplest form does not have a radical in the denominator.
For a fraction of the form √ba where a∈Z,b∈N, we find the simplest form by mutliplying the numerator and denominator by √b:
√ba=ba√b.
When we remove a radical from a denominator, we call it rationalizing the denominator.
Rationalizing Denominators with Conjugates
SL AA 1.7
To simplify a fraction of the form b+√ca, multiply the fraction by b−√cb−√c.
b−√c is called the conjugate of b+√c.
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