# 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.

**Example 1**

, n odd. Examples:

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

**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]
.

Hint: Rewrite the equation
, using fractional exponents.

Step 1:

Hint: Make a table of x, y values that satisfy the
equation Use x -values

−2, − 1, 0,1, 2 and round to the nearest hundredth.

Step 2:

Hint: Plot these five points you just found.

Step 3:

Hint: Smoothly connect the dots.

Answer:

**Chapter 6 Section 2 Lesson: Radicals and Rational
Exponents**

Extended Example 3b

Graph the function over the interval [−2, 2]
.

Hint: Rewrite the equation , using fractional
exponents.

Step 1:

Hint: Make a table of x, y values that satisfy the
equation . Use x -values

and round to the nearest hundredth.

Step 2:

Intermediate

calculation

Hint: Plot these eight points you just found.

Step 3:

Hint: Smoothly connect the dots.

**Chapter 6 Section 2 Lesson: Radicals and Rational
Exponents**

Extended Example 3b, continued…

**Extended Example 3c**

Graph the function over the interval [−1, 5]
.

Hint: Rewrite the equation , using fractional
exponents.

Step 1:

Hint: Make a table of x, y values that satisfy the
equation. Use x -values

and round to the nearest hundredth.

Step 2:

**Chapter 6 Section 2 Lesson: Radicals and Rational
Exponents**

Extended Example 3c, continued…

Hint: Plot these eight points you just found.

Step 3:

Hint: Smoothly connect the dots.

Answer:

End of Lesson