Power of a fraction
"To raise a fraction to a power, raise the numerator
Solution. We must take the 4th power of everything. But to take a power of a power -- multiply the exponents:
Problem 1. Apply the rules of exponents.
To see the answer, pass your mouse over the colored area.
In the previous Lesson we saw the following rule for reducing a fraction:
If the numerator and denominator have a common factor,
Consider these examples of canceling:
If we write these examples with exponents, then
In each case, we subtract the exponents. But when the exponent in the denominator is larger, we write 1-over their difference.
Here is the rule:
Problem 2. Simplify the following. (Do not write a negative exponent.)
Problem 3. Simplify each of the following. Then calculate each number.
Solution. Consider each element in turn:
Problem 4. Simplify by reducing to lowest terms. (Do not write negative exponents.
We are now going to extend the meaning of an exponent to more than just a positive integer. We will do that in such a way that the usual rules of exponents will hold. That is, we will want the following rules to hold for any exponents: positive, negative, 0 -- even fractions.
We begin by defining a number with a negative exponent.
It is the reciprocal of that number with a positive exponent.
a−n is the reciprocal of an.
The base, 2, does not change. The negative exponent becomes positive -- in the denominator.
Example 6. Compare the following. That is, evaluate each one:
3−2 −3−2 (−3)−2 (−3)−3
−3−2 is the negative of 3−2. (See Lesson 13.) The base is still 3.
As for (−3)−2, the parentheses indicate that the base is −3:
A negative exponent, then, does not produce a negative number. Only a negative base can do that. And then the exponent must be odd
Solution. Since we have invented negative exponents, we can now subtract any exponents as follows:
We now have the following rule for any exponents m, n:
In fact, it was because we wanted that rule to hold that we
a−1 is now a symbol for the reciprocal, or multiplicative inverse, of any number a. It appears in the following rule (Lesson 5):
a· a−1 = 1
Problem 5. Evaluate the following.
Example 9. Use the rules of exponents to evaluate (2−3· 104)−2.
Problem 7. Evaluate the following.
g) (½)−1 = 2. 2 is the reciprocal of ½.
Problem 8. Use the rules of exponents to evaluate the following.
Problem 9. Rewrite without a denominator.
Example 10. Rewrite without a denominator and evaluate:
Answer. The rule for subtracting exponents --
-- holds even when an exponent is negative.
Exponent 2 goes into the numerator as −2; exponent −4 goes there as +4.
Problem 10. Rewrite without a denominator and evaluate.
The reciprocal of a−n.
Reciprocals come in pairs. The reciprocal of an is a−n :
And the reciprocal of a−n is an :
Factors may be shifted between the denominator and the numerator
The exponent 3 goes into the numerator as −3; the exponent −4 goes there as +4.
Problem 11. Rewrite with positive exponents only.
Problem 12. Apply the rules of exponents, then rewrite with positve exponents.
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Copyright © 2012 Lawrence Spector
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