Pure Mathematics 10 - Unit One - Real Numbers
Pure Mathematics 10 - Unit One - Real Numbers
Lesson Nine:
- 1.8 Rational Exponents (pp. 34 - 38)
- exponents with numerator = 1
- exponents with numerator = m
Objectives:
- Students will:
1) evaluate expressions with rational exponents
1) Rational Exponents
- we know what positive exponents like 23 or 34 mean and we should know what negative exponents mean. For example:
- what, however, do we get when we want to find a fractional (rational) exponent like:
- to narrow down our search, let us look at some things that we should already know. For example, 30 = 1 and 31 = 3. It stands to reason that, because:
- therefore, without knowing the actual value of 3½ we know that it should be between 1 and 3
- this type of explanation is used quite often in mathematics and is known as deductive reasoning
Notes:
- we know from our multiplicative exponent law that:
- if we choose to multiply 3½ by itself we get:
- we also know from experience that:
(if you are not sure that this is true, substitute any natural number in place of the "a" in the above statement and see what you get)
- let's say that we take the square root of 3 or:
- notice the similarities between the two equations:
- because the two situations are both equal to 3 we can merge them together or say that:
- it would appear (although we can not actually say that we have "proved" this) that:
Another Proof (optional)
- this "proof" is taken from p. 34 of Mathpower 10 and may be done on the students' own time if so inclined:
- by using the power law for exponents, we can say that 9 can be written as:
because:
and:
- In general, we can say that:
G.P. Evaluate
Ans:
Thus, the formula appears to work for positive and negative radicands and for positive and negative exponents
**N.B.** this should be a review from lesson three, but in case it is not:
index
radicand
radical sign
- what happens when our exponent is not equal to 1?
- let’s take, for example:
- we know that:
- similarly, we know that:
- so it would reason that if we squared both sides of the equation:
- and then used the distributive property to distribute the exponent:
- we can say that:
In general, for m ( I, n ( N:
where, if n is even a ( 0
Another Proof:
- this “proof” is taken from p. 36 of Mathpower 10
- using the power law we know that:
- where, if no index is given, it is assumed to be 2
G.P. Evaluate the following:
Ans:
G.P. Simplify.
Ans:
Assign: pp. 37 – 38 # 1 - 48 (every third question)
# 49 - 83 (every second question)
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