CHAPTER 6 - EXPONENTS AND RADICALS - (Pages 277 to 324) In Chapter One, the notation
xn = x(x)x(x)x … x, n - x's
was introduced where the positive integer n is called the exponent, and x is called the base. A number of basic properties of exponents were also discussed. In this chapter we will expand n to include the set of rational numbers.
LESSON #6.1 - The General objective of this lesson is to show how the concept of exponents can be expanded to include integers.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to multiply and divide expressions involving integer exponents.
WORD PROBLEMS:
6.1 - Using Integers as Exponents - In Chapter Four it was shown that when n is a natural number, we had the following properties for exponents.
Properties of Natural Number Exponents - If x and y are real numbers, and m and n are natural numbers. then:
Consider Propery 1:
xmxn = xm+n
If we let n =0, this becomes -
xmx0 = xm
For internal consistency we need the following definition.
Definition of the Zero Exponent - If x is any non zero real number, then-
x0 = 1.
Again, starting with Propery 1:
xmxn = xm+n
If we let n = -n, we get-
x-nxn = x0 = 1.
Again, for internal consistency we need the following definition.
Definition of Negative Integer Exponents - If x is any non zero real number, then-
x-n = 1/xn
With these two definitions we can expand our rules for exponents to include the set of integers.
Properties of Integer Exponents - If x and y are real numbers, and m and n are integers, then:
Board and Class Problems - Page 283/12, 22, 40, 52, 72, 80.
LESSON #6.2 - The General objective of this lesson is to introduce roots and radicals.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to express radical expressions in simplest form.
6.2 - Roots and Radicals - We made the jump from natural to integer exponents with little difficulty. The case, however, is different when we consider rational exponents, because of the rise of apparent ambiguities. For instance, if we want to find the square of the number 2, we write
22 = 4
The inverse of this process is called for finding the square root of 4, and then we write
Let us now consider the following situation. We can write that
(-2)2 = 4.
Therefore, we can certainly say that another square root of 4 is - 2. Does this means that we can write that
The answer to this question is no! However, we can write that
This means that we have to distinguish between the different roots of a number. In fact, if we restrict ourselves to the real number system, many of these roots do not exist.
The Principal nth Root of a Number - If a and b are in the set of real numbers, and n is a natural number greater than one, then if -
an = b
we can write that -
where -
The entire expression is referred to as a radical.
The principal nth root of b satisfies the following conditions.
1. If b>0, then the principal nth root of b is the positive root of b.
2. If b<0 and n is odd, then the principal nth root of b is the negative root of b.
3. If b<0 and n is even, then the principal nth root of b is not defined.
4. If b = 0, then the principal nth root of b is equal to zero.
Let us reconsider the problem of finding the square root of four. As we saw there were two answers. One really has no priority over the other, and unless you are specifically asked for one or the other, you should always respond plus and minus two to the question;
What is the square root of four?
In general, the number of roots a number has is equal to the index.
Radicals- The word radical comes from a Latin word radix, meaning root. A German mathematician named Christoff Rudolff who first used it in 1525 invented it. As a result of the previous definitions we can list of the following properties for radicals.
Properties of Radicals- Let x and y be real numbers, and n is a natural number greater than one. Then if all expressions are defined as real numbers, we can write;
Simplest Radical Form - A radical expression is in its simplest form when it satisfies the following conditions.
Board and Class Problems- Pages 293 to 294/12, 20, 24, 46, 64, 76, 86a,b,e.
LESSON #6.3 - The General objective of this lesson is to learn how to add and subtract radical expressions.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to;
6.3 - Combining Radicals And Simplifying Radicals That Contain The Variables- Expressions such as
are called radical expressions.
Addition and Subtraction- When adding or subtracting radical expressions, we must combine only like terms. This means that the terms must have the same index and the same radicand.
Board and Class Problems - Pages 298 to 299/10, 20, 28, 54, 66, 72.
LESSON #6.4 - The General objective of this lesson is to learn how to multiply and divide using radicals.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to multiply and divide radical expressions.
6.4 Products and Quotients Involving Radicals- When multiplying on dividing radical expressions we have much more leeway. However, is usually preferable not to leave radicals in the denominator. Rationalizing the denominator, which utilizes the special products, does this.
x2 - y2 = (x + y)(x - y)
x3 - y3 = (x - y)(x2 + xy + y2)
Multiplication/Division- To multiply and divide expressions involving radicals we use the falling two properties.
Board and Class Problems - Page 304/14, 20, 26, 38, 42, 44, 50, 60, 74.
LESSON #6.5 - The General Objective of this lesson is to learn how to solve radical equations..
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to solve equations involving radicals
6.5 - Equations Involving Radicals- Equations are sometimes written using radical notation. For instance
is a radical equation. The procedure for solving equations of this type depends upon the following property.
Property- Let x and y be real numbers and let n be a natural number.
If x = y , then xn = yn .
Procedure for solving radical equations.
Step 1 - Isolate one of the radical expressions containing the variable on one side of the equation.
Step 2 - Eliminate the radical by raising both sides of the equation to a power equal to the index of the radical.
x + 8 = x2 - 8x + 16
Step 3 - Repeat the first two steps as often as necessary, in order to eliminate all radicals.
Step 4 - Solve the equation in Step 3.
x2 - 9x + 8 = 0
(x-8)(x-1) = 0
x = 8 and x = 1
Step 5 - Check all solutions obtained in the original equation. Discard any extraneous solutions, which may be introduced whenever a radical with an even index is involved.
The solution is -
x = 8.
The apparent solution of -
x = 1
is an extraneous solution, and should be discarded.
Whenever a radical equation involves an even index, it is always possible to introduce extraneous solutions. The only guarantee we have, is that if the original equation has a solution, it is included in the solution set of the new equation where the radicals have been eliminated.
Board and Class Problems- Pages 309 to 310/24, 28, 46.
LESSON #6.6 - The General objective of this lesson to show how exponents and roots are related.
PERFORMANCE OBJECTIVES - At the end of this lesson the student will be able to convert from exponential notation to radical notation, and vice versa.
6.6 - Merging Exponents and Roots-we are now ready to show how exponential and radical notation or related. Recall that-
If the exponential rule -
(xm)n = xmn
is to hold when m is a rational number of the form, m = 1/n, then since -
(x1/n)n = x
it is necessary that exponents and radicals be connected by the following rule -
Rational Exponents- Let x be a real number not equal to one, and m,n are natural numbers. If the fraction m/n is reduced to lowest terms, and if the indicated radicals exist, we can write -
Properties Of Rational Exponents- Let x and y be real numbers, and p and q are reduced rational numbers. Then if expressions are defined as real numbers, we can write-
1. xpxq = xp+q
2. (xp)q = xpq
3. (xy)p = xpyp
4. (x/y)p = xp/yp
It often becomes confusing when working with radical notation. If you have trouble with it, you might find it easier to convert from radical to exponential notation, carrying out the operations in exponential form, and then convert the final result back to radical form.
Board and Class Problems - Pages 314 to 315/6, 10, 16, 18, 38, 44, 60, 78.
HOMEWORK - Lesson Six Assignment
COMMENTS -