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Radicals and Rational Exponents

Introduction

Radicals are really about dividing exponents. In this section, we’ll see how that idea leads us to fractional
exponents, and how all of the usual rules for exponents apply even when the exponents are fractions. This
concept gets to the heart of radicals and is the key to solving many problems involving radicals. In fact,
you need to know how radicals are equivalent to fractional exponents in order to evaluate radicals (other
than square roots) on a typical scientific calculator.

Although you already know that the square root of nine is three, focus carefully on the new way of
looking at this simple square root problem illustrated in the following example.

How does taking the square root of nine involve dividing exponents? It turns out to be a matter of “two
goes into two once with no remainder”:

Not only is the radical symbol similar to the long division symbol , it really does indicate
division—division of exponents.

Example A involves a fifth-root radical expression, and it has a remainder.

Example A

Simplify:

We’ll come back to this example a little later, to look at each step in depth. For now, just watch
how the overall process unfolds…

“5 goes into 23, 4 times with a remainder of 3”.

Notice that the remainder actually does remain under the radical.

Considering this example, it makes sense to write:

More generally, an n^th root divides exponents by n , so we can write:

That last equation is a bridge between radicals and exponents. Crossing this bridge gives us
access to the powerful exponent properties that you know (and love) so well…but if you don’t
know them quite so well, this may be a good time to review the Properties of Exponents (see the
table on the next page).

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Properties of Exponents

Now let's redo Example A, but just using exponents.

Example B
Simplify:

Above, we learned thatIt can help to see how this is true by noting that as a
mixed number is which means

See how this relates to the process below (each exponent rule used is referred to by its Roman
numeral in the Properties of Exponents table):

Using a Calculator to Find Roots

How do we find a numerical approximation to expressions like using a standard scientific
calculator? Calculators have square root buttons but no fifth root buttons, so what do we do? Fractional
exponents, to the rescue!

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Example C
Evaluate:

The word “evaluate” here implies that we want a numerical approximation to this radical. So, go
get your calculator.

Scientific calculators allow you to raise numbers to powers. Try to compute this on your
calculator. On some calculators you enter:

On others, including the one mentioned in the Introduction to this section, enter:

If you have a “reverse-Polish” calculator, enter:

Note: Some calculators have an button not . They do the same thing.

You should get something close to 2.634879412770604848540719 (though probably with
fewer digits).

It’s very important that you know how to compute radicals on your calculator. Consult your
calculator’s manual if none of the methods above work for you.

Extended Example 1a
Approximate to the nearest millionth.
Hint: Rewrite, using fractional exponents.
Step 1:

Hint: Compute this on your calculator.
Step 2:
6.77372038951128 (Your result may vary, slightly!)
Hint: Round your result to the nearest millionth.
Answer:
6.77372038951128
6.773720

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Extended Example 1b
Approximate to the nearest millionth.
Hint: Rewrite, using fractional exponents.
Step 1:

Hint: Compute this on your calculator.
Step 2:
9.96491197718962 (Your result may vary, slightly!)
Hint: Round your result to the nearest millionth.

Answer:
9.96491197718962
9.964912

Extended Example 1c

Approximate to the nearest millionth.
Hint: Rewrite, using fractional exponents.
Step 1:

Hint: Compute this on your calculator.
Step 2:
4.75154626941184 (Your result may vary, slightly!)
Hint: Round your result to the nearest millionth.
Answer:
4.75154626941184
4.751546

Example D

Approximate to the nearest millionth.

First we rewrite the radical, using fractional exponents:

Notice how the root (the index) becomes the denominator, and the exponent under the radical
becomes the numerator.

Computing this on my calculator, I get:
20.4239239445 (Your result may vary, slightly!)

Rounding to the nearest millionth, I obtain:

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Extended Example 2a
Approximate to the nearest hundred-thousandth.
Hint: Rewrite, using fractional exponents.
Step 1:

Hint: Compute this on your calculator.
Step 2:
41.3543377040440 (Your result may vary, slightly!)
Hint: Round your result to the nearest hundred-thousandth.
Answer:

Extended Example 2b
Approximate to the nearest hundred-thousandth.
Hint: Rewrite, using fractional exponents.
Step 1:

Hint: Compute this on your calculator.
Step 2:
215.611017534756 (Your result may vary, slightly!)
Hint: Round your result to the nearest hundred-thousandth.
Answer:

Extended Example 2c
Approximate to the nearest hundred-thousandth.
Hint: Rewrite, using fractional exponents.
Step 1:

Hint: Compute this on your calculator.
Step 2:
7413.00053725752 (Your result may vary, slightly!)
Hint: Round your result to the nearest hundred-thousandth.
Answer:

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Next we'll see how algebraic properties of radicals can be understood in terms of the familiar properties of
exponents. But first, study the following properties of radicals.

Properties of Radicals (for n = 1, 2,... )		Examples
I	For n odd or A ≥ 0 and B ≥ 0 .	See Example 1 below
II	For n odd or A ≥ 0 and B ≥ 0 , and B ≠ 0 .	See Example 2
III	For n odd or A ≥ 0 .	See Example 3
IV	For n even.	See Example 4
V	For n odd or A ≥ 0 .	See Example 5
VI	For all n .	Not applicable
VII	For all n .	Not applicable

Example 1
, n odd. Examples:

**This step is true because (using this rule in reverse):

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

continued…

Example 2

As usual, we must always have nonzero denominators, since
division by zero is undefined.

Example 3

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

continued…

Example 4

Example 5

Example E

Approximate to the nearest millionth.

For odd roots, a negative under the radical can come out of the radical, as we saw in the previous
section:

Note that this is possible because (using the property of radicals shown in the
table above):

Next we rewrite the radical, using fractional exponents:

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Example E, continued…

Notice how the index becomes the denominator, and the exponent of the radicand becomes the
numerator.

Calculate on your calculator, and don’t forget to keep the negative in front of your final
answer:
−105.825353941 (Your result may vary, slightly.)
Rounding off to the nearest millionth, we get:

Now we'll see how the properties of radicals arise from the properties of exponents.

Example F
Translate the leftmost and rightmost of the sequence of equalities below into radical notation:

**Note: This is a valid step, due to the following property of exponents:

Since, (because), we have an important property of exponents:

This leads us to this property of radicals:

Recall that this is true only when A is non-negative. So, to be well-defined, only non-negative
numbers may be raised to rational powers with an even denominator (reduced to lowest terms).

Important Note
In this course you can assume the appropriate
conditions are met when doing algebra with
exponents. From now on, you can assume that all
variables represent positive quantities unless it is
clearly stated that they do not.

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Example G
Translate the leftmost and rightmost of the sequence of equalities below into radical notation:

**Note: This is a valid step, due to the following property of exponents:

Substituting we get

This result, , is radical property V in the Properties of Radicals table. Recall that the
result of Example F was the property which is also property V, but with n = 2 :

(Don’t forget that an index of 2 is always implied for square roots:

Example H
Translate the following equation into radical notation:

**Note: This is a valid step, due to the following property of exponents:

Substituting we get:

This is radical property I in the Properties of Radicals table.

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Graphing Radical Functions using Fractional Exponents
Now we’ll see how to graph functions by using fractional exponents to calculate the points to plot.

Example I
Graph the function over the interval [−6,6] .

To graph H , first rewrite it using fractional exponents:

Next we make a table of x, y values that satisfy the equation(using a calculator). In the
table, all values are rounded to the nearest hundredth:

Then we plot the points found:

Be on the lookout for “regions of uncertainty,” where you need to plot additional points. This
happens whenever the change in a graph is abrupt, and you aren’t sure where the graph should go,
as in the region indicated by question marks in the graph above.

We’ll plot two more points, at x = −0.25 and x = 0.25 , to better see how the curve passes
through the region of uncertainty.

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Example I, continued…

Adding these two points to our plot, we can then connect the dots:

Points like (0,0) in the graph above are called cusps.
Note: There is an animation of this example in the course online

Example J
Graph the function over the interval [0,6].

First, rewrite the function using fractional exponents.

Make a table of x, y values that satisfy the equation

Use x -values 0,1, 2,3, 4,5,6 and round to the nearest hundredth.

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Example J, continued…

Plot the seven points we found

Find and plot two more points, say at x = 2.5 and 3.5, to better see where the curve passes
through the regions of uncertainty (indicated with question marks).

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Example J, continued…

Plot the two additional points.

Note: The graph shown is a precise graph of the function, done by a computer program. To approach this
accuracy by hand, it would be necessary to plot at least two additional points, say at x = 2.75 and at
x = 3.25 .

Chapter 6 Section 2 Lesson: Radicals and Rational Exponents

Extended Example 3a
Graph the function over the interval [−2, 2] .