§7.1 & 7.2 Radicals and Radical Functions & Rational Exponents
a = bb taken n times is equivalent to a
"n" is the index
"a" is the radicand
is the radical
"b" is the root
Note: A radical expression looks for the base"b" that when raised to the power of the index, yields the radicand.
Some Basics
1) If the index is even then the radicand can't be negative
Example: -4 = Not a since no number, times itself, an even
number of times yields a negative number.
2) If the index is odd then the radicand can be negative and if so the root is negative
Example: -8 = -2since a negative number times itself an odd
number of times yields a negative number.
3) Every even indexed radical has 2 possible answers, the principle root (non-negative/positive root) and the secondary root (negative root)
Example:√4 = ± 2
At this time, if you don't already have them memorized, I highly recommend the memorization of the following perfect squares and perfect cubes!
Perfect Squares 1-15 and 25
12=1, 22=4, 32=9, 42=16, 52=25, 62=36, 72=49, 82=64, 92=81,102=100, 112=121, 122=144, 132=169, 142=196, 152=225, 252=625
Perfect Cubes 1-5
13=1, 23=8, 33=27, 43=64, 53=125, 63 =216, 73 =343
Example:Evaluate
a)1/4Square root of quotient is quotient of square roots
b)-0.125Deal with the # and ignore the decimal until the end
c)- 16Break down to primes and divide by index to get root
d)√ -25Even index & negative radicand No Real Solution
d)15/13Round the answer to the nearest 1000th from calculator
Taking the roots of variable expressions is even easier than taking the roots of numbers, but it is a nice segway into rational exponents. The process is very simple and the reason is rational exponents which we will get to in a moment.
Process for Root of a Variable Expression
1) Divide the exponent of the variable by the index
2) Rewrite without the radical symbol, as variable to quotient
Example:Find the root. Assume the variable represents a positive number.
a)x4y2b) x16y32
c) x9y12z21d)√256x4y6
d) 343x36y15
Note: We’ll find out in the next section what happens when we have remainders.
A radical expression can also be written as an exponent. I always teach it from the get go, as it leads to less confusion in the end! It also gives us a chance to see it several times before the class is over.
A rational (fraction) exponent represents a radical expression. One over the index represents the radical expression xm .
xm = x equivalent to the nth root of "xm"
Note: This is useful when finding the nth root of a very large number and for applications of radical expressions. This is actually what we used when simplifying a root containing a variable expression.
Let’s just practice the conversion between radical expressions and rational exponents first.
Example:Write each in exponential form.
a)3 x 17b) 4 x13
c)2 x 13d)( 2 x )13
Note: It doesn’t matter whether the exponent is outside or inside the representation appears the same when dealing with positive numbers! Remember that we assume the variables are positive unless otherwise stated. Next, we will deal with some special circumstances that have to do with the value of the expression.
c)3 4x + 3y2
So, as I brought up in the last note, if you do not know that the variable expression represents a positive number, then we must use absolute values, to represent the root when the index and exponent of the radicand are both even. Here is the summary and example by way of explanation.
For even index & power of radicand:n√ an = | a |
For odd index & power of radicand:n√ an = a
Example:Find the root of 4√(-3)44√(3)4
Note: In the first you’ll eliminate the negative by taking it to an even power, so the root will exist and will end up being the 3, which is the same as the second problem. This is the catch, based on an answer, you will never know whether the base in the radicand started out as a positive or a negative, so when we don’t have any pre-knowledge, we say that it can be either positive or negative. This is actually an important fact in solving radical equations, but that connection is generally left out!
Example:Find the root of each (don’t assume that the variable is positive)
a)√ (-5)2b)6√(x – 6)6
c)3√(-3)3d)5√(7 + 2x)5
e)√16x2f)√9x2 + 6x + 1
Example:Write each expression in radical form
a)ab)x
c)(5x2)d)(2x y2)
e)– x
Writing a radical expression in its equivalent rational form allows us to more easily evaluate radical expressions containing variables and even large numbers. The trick is to re-write numbers using their prime factorization in exponential form and then to deal with the numbers just as we did with the variables. We will assume that variables are non-negative unless otherwise specified.
All of the exponent rules that you have learned apply to rational exponents, and this only increases the power of the rational exponent. Let's review:
Multiply like bases – Add exponents
x x = x
Divide like bases – Subtract denominator exponent from numerator exponent
x = x
x
Negative Exponent – Take the reciprocal of the base
x = 1
x
Power Rules – Multiply the exponents
( x ) = x
(x2 y4 ) = x y
x2 = x
y4y
Now we can use our skills in rewriting a radical expression into rational exponents, in conjunction with our exponent rules to simplify all manner of radical expressions!
Example:Simplify by changing to exponential form. Write the
answer in radical form when appropriate.
a)3 x 12b)12 x 3
c)( 3x2yz3 )21d)(4x2yz3 )2
e) x4f)3 4 x
We’ll also want to be prepared for the radicals that are already expressed as rational exponents, and be able to evaluate them as if they were already written with a radical symbol.
Example:Evaluate if possible. If the expression isn’t , so state.
a)(-125)b)(-100)
c)(-27)d)16 + 25
Example:Write in exponential form without negative exponents.
a)a ab)x x
c)( x )d) x
x
e) a
a
Note: On these problems it is the exponent rules that are being used. What makes that cumbersome is that the rules are being used on fractions which means that we must add and subtract fractions when we are multiplying and dividing like bases.
This is a skill that we can now take and use to simplify radical expressions that are multiplied and divided, but have different indexes.
Process for Rewriting
1) Rewrite as rational exponents (if a number, use prime factorization first)
2) Use exponent rules on like bases
a)Simplify the resulting fraction addition or subtraction problem
3) If unlike bases find LCD of denominators (indexes) & build higher terms for all rational
exponents
4) Rewrite in radical form
Example:Rewrite the following radical expression into a single radical
expression using your knowledge of rational exponents and
exponent rules.
a) 4 x b)3y 4x
5 x
a) 3 x2 b)43 34
6 x2
Now, a little miscellaneous wrap-up and a preview of some of chapter 2’s lectures. We have a special way of writing evaluation problems in math and science, one that incorporates both a variable representation of the input (independent values) and the output (dependent values) of an equation in 2 variables. This special way of writing evaluation problems is called function notation. It looks like an algebraic expression set equal to a variable with parentheses containing the value to input, and it says, find the value of the variable (read as function) when the existing variable in the algebraic expression is equal to the value in the parentheses. This is what it looks like:
f(x) = x2 + x + 2
Here is an example of an evaluation problem where we would evaluate the algebraic expression on the right of the function above.
Example:Evaluate x2 + x + 2 when x = -3
Now, here is the same problem given using function notation.
Example:Findf(-3) forf(x) = x2 + x + 2
Now, the relationship to our work.
Example:For the functionsf(x) = √x + 2g(x) = 3√ x – 1 find
a)f(7)b)g(9)c)f(0)d)g(10)
Y. ButterworthCh. 7 Notes – Martin-Gay/Greene1