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.
Dividing powers of the same base
In the next Lesson we will see the following rule for reducing a fraction:
"Both the numerator and denominator may be divided
Consider these examples:
If we write those with exponents, then
In each case, we subtract the exponents. But when the exponent in the denominator is larger, we write 1-over the 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 6):
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.
Problem 11. Factors may be shifted between the numerator and denominator [How?]
By changing the sign of the exponent.
The exponent 3 goes into the numerator as −3; the exponent −4 goes there as +4.
Problem 12. Rewrite with positive exponents only.
Problem 13. Apply the rules of exponents, then rewrite with positve exponents.
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