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Discussion
So far, we have only seen integer exponents, but fractional exponents exist too. Using your knowledge of exponential laws, determine the value of 6431.
Basically, we have found that 6431 is the number whose cube is 64.
Using the same thinking, we can see that 221 is the number whose square is 2, but this is exactly the definition of a square root!
You have probably seen a square root with radical (√) notation before but there are other kinds of roots! We typically write them in radical form as:
The radical notation for a square root is a shorthand since the square root is the most common root, but we can write square roots with the more general notation too:
Returning to our original example, now we can say:
We call this the cube root of 64, or the number whose cube is 64.
nth Roots
For any number a and whole number n,
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:
Note that this root is positive when n is even.
Checkpoint
5√6 raised to what power equals 6?
Select the correct option
Discussion
Does −64 have a real square root? How about a cube root?
Roots of negative numbers
If a is negative, n√a is negative for all odd n.
For even n, no real n√a exists.
Checkpoint
Evaluate 3√−27.
Select the correct option
Recall that we introduced the roots 3√64 and √2 as fractional exponents. We can do this with any nth root!
Converting nth roots to fractional exponents
Roots can always be written as fractional exponents and vice versa:
Checkpoint
Write 8√26 as an exponent.
Select the correct option
Discussion
Now, we can understand fractional exponents of the form an1 through our knowledge of nth roots! However, can we interpret other rational exponents, like 52?
Utilizing your knowledge of nth roots and exponential laws, determine the value of 3252.
Rational exponents
Utilizing nth roots and exponential laws we can rewrite any rational exponent:
Exercise
Write 7√x61 in the form xa/b.
Select the correct option
Exercise
Evaluate 12532.
Select the correct option
Now, that we understand what nth roots are, we can learn about their algebraic properties.
First, since roots are a type of exponent, all of our exponent rules carry over to roots. For example, we learned that
so we can see that
Understanding that
we can learn how to put radicals into simplest form.
Simplest form radicals
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:
Checkpoint
Simplify 4√96.
Select the correct option
Exercise
Simplify as much as possible (3√272√3⋅√12)3.
Select the correct option
We can also simplify fractions containing radicals:
Simplest form fractions with radicals (multiplying by roots)
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:
When we remove a radical from a denominator, we call it rationalizing the denominator.
Checkpoint
Simplify √21.
Select the correct option
Discussion
However, not every fraction with a radical is of the form √ba.
Evaluate 1+√31⋅√3√3. Explain whether your result is a fraction in simplest form.
We see that our old strategy of multiplying by a radical does not work if the denominator has another term!
The core issue we run into is that there is no nonzero number we can multiply both 1 and √3 so that both products are rational. Instead, we need to multiply the numerator and denominator of a fraction like
by a quantity with multiple terms.
Discussion
Using the fact that (a+b)(a−b)=a2−b2, can you find a nonzero quantity x such that (1+√3)x contains no radicals?
Now, we know how to simplify 1+√31:
We multiplied the numerator and denominator by the conjugate of 1+√3.
Rationalizing Denominators with Conjugates
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.
Exercise
Rationalize the denominator in the fraction 3−√22 and simplify.
Select the correct option
Exercise
Simplify 16−1(423⋅2−7⋅831)2.
Select the correct option
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