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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