Radical Expressions and Functions

1
Radical Expressions and Functions

• Section 8.1
• MATH 116-460
• Mr. Keltner

2
Finding the nth root of a number

• Finding the square root of a number involves
  finding a number that, when squared, equals the
  given number.
• In other words, finding such
  that b2 a.
• Some vocabulary involved with nth roots

This is called a radical symbol.
n is the index of the expression. The index tells
us what amount of factors we should look for in
order to simplify a quantity. Examples If n 3,
we are looking for some value r such that r3
s. If n 4, we are looking for some value r such
that r4 s.
s is called the radicand of the
radical expression. If the index n is even, then
s must be positive. This is because there is
no value of r such that r2 -s.
3
Nth roots generally speaking

• We can define the nth root of a number as this
• b is the nth root of a number a if bn a.
• The principal root of a nonnegative number is
  just the nonnegative root.
• Example The square root of 16 could be 4 or -4,
  but it is easiest to find a single root, so we
  usually only indicate the principal root.
• The only exception would be solving an equation,
  where either the principal root or the secondary
  root could both supply solutions to the equation
  (that is why it becomes important to check your
  answers).

4
Evaluating nth roots

• When evaluating a radical expression like
  the sign of a and the index n will determine
  possible outcomes.
• If a is nonnegative
• Then , where b 0 and bn a.
• If a is negative
• And n is even, then there is no real-number root
  (no real solutions). We will cover these
  expressions in section 8.7.
• And n is odd, then , where b is
  negative and bn a.

5
Example 1 Evaluating Nth roots

• Simplify each of the following roots.
• Click each expression to reveal the answer.

-2
0.6
13
No real root
5 6
4
-3
-3
6
Irrational Roots

• All the examples we have seen so far have been
  rational solutions (those that are whole numbers
  or decimals that terminate or have some repeating
  pattern, like 1/3 or 2/11.
• Some roots, like , are called irrational,
  because they do not have a decimal that ends or
  has a repeating pattern.
• Another example of an irrational number is p,
  which we usually just round to 3.14.
• We cannot express the exact value of this
  expression without rounding error of some sort.
• So, is actually considered the exact value
  of .

7
Evaluating Roots with a Calculator

• When we graphed polynomial functions, it became
  evident that our graph was more accurate by
  plotting additional points on the graph.
• When evaluating roots on a calculator, our answer
  becomes more accurate with more decimal places.
• This table illustrates how accurate our estimate
  of becomes by using more and more decimal places.

8
Nth Roots on a Calculator

• Evaluating nth roots on a calculator can be easy,
  but you must know where to look.
• Using the MATH key (just below the ALPHA key),
  you can evaluate any nth root you wish and the
  calculator will return the principal root.
  Follow these steps
• Enter the index of the expression,
• Press the MATH key,
• Then 5xv( and enter the value you want to
  evaluate.
• These steps are very similar for several models
  of calculators. Try these

9
Simplifying Radical Expressions

• Definition of roots

• Power of a Power

• Remember that, when the index is even, the
  principal root will be nonnegative.
• This allows us to assume that all variables are
  represented by nonnegative values.

• The Power of a Power property of exponents tells
  us that (am)n amn.
• We will use this property to verify roots in the
  next examples, as well as to help us simplify
  irrational roots.
• By writing an original quantity as a perfect
  square, it may help simplify the square root of
  the quantity.

10
Example 3 Simplifying Expressions

• Simplify each of the following roots.
• Click each expression to reveal the answer.

3x5
x2
b2
6a6
3y3
11
Radical Functions

• Radical functions are those that are defined by a
  radical expression.
• From section 3.1, we will see them in function
  notation, usually as f(x) or similar notation.
• This is, essentially, a fancier way of writing
  y.
• To evaluate for a particular value of x, such as
  2, the notation will simply look like f(2).
• We do not have to divide the equation when we are
  done, just substitute the given value for the
  variable and simplify.

12
Domain of Radical Functions

• With even indexes, it is not possible to evaluate
  the root of a negative value.
• Because of this, we must only use input values
  that result in nonnegative outputs.
• To evaluate the domain in such cases, set the
  radicand greater than or equal to zero (0) and
  solve to get the domain of the function.

13
Example 4 Domain of Radical Functions

• Evaluate the domain of each radical function.
• Click the function to see the answer.

-2 x 2
x 4
14
Assessment

• Pgs. 541-543
• s 9-108, multiples of 3
• (most are calculator problems)