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We will now introduce how to work with negative exponents. Before we give a definition, let’s experiment with our calculators. 3Evaluate: 2
Directions:
Type 2 then the exponent button. The exponent button will either be xy or .Then type in –3. Use the (+-) button to make the 3 negative. You should see 0.125 on the calculator.
0.12518
Notice the result of a positive number with a negative exponent. The result is not a negative number.
2Try another: Evaluate: 50.04
125
Most calculators can convert from decimal to fraction. Some calculators, you can just press the fraction button then enter. On others, the DF button will convert the decimal to a fraction.
3 21 1So 2 and 5 .
8 25 You may notice, the result is the base to
the positive exponent under one.
nn
If n is apositive integer and b is a real number (b 0), then:
Definition: If b is any nonzero real number, b0=1.
Example: 70=1 Try using a scientific calculator to verify this is true. If you don’t have one, then get one. I’m sure your instructor would be more than happy to recommend an appropriate calculator.
More Examples:
015 1 03
111
0n 1 03 4x y 1 , ,,
This definition is consistent with property 5i) from the previous slide.
If we subtract the exponents, the exponent equals zero.
We’ll now simplify a quotient of two variables with negative exponents. An expression is considered simplified if it does not contain negative exponents and like variables are combined using properties of exponents.
Simplify: 3
5
x
y
(Shortcuts given after example.)
-nn
11. Use the property b = to rewrite without
b negative exponents.
2. Simplify the complex fraction.
3
5
1
x1
y
3 5
1 1
x y
5
3
1 y1x
5
3
y
x
Shortcut: If the exponent is negative, move it from denominator to numerator or numerator to denominator, then change the exponent to positive. If the exponent is positive, leave it where it is.
Example 2. Simplify the following. Use the shortcut:
7
2x
a) y
4
9x
b) y
7
1x
c) y
Solutions:
Shortcut Procedure: If the exponent is negative, move it from denominator to numerator or numerator to denominator, then change the exponent to positive. If the exponent is positive, leave it where it is.
Now that we know how to deal with negative integer exponents, we will be able to simplify more types of problems. Please note: There will usually be more than one method of simplifying these expressions. Recommendations will be given on the “easiest” method, however, it certainly will not be the only method. Recall the properties of exponents and the definition of a negative exponent:
n
n mm
b 5i) b if n m
b
n
m m nb 1
5ii) if n<mb b
nn m
mb
b .b
We can rewrite #5 simply as:
Next Slide
Then use the definition of a negative exponent if necessary:
It is definitely a good idea to apply the exponent outside the parentheses to all of the exponents inside. Be careful, if a variable doesn’t have an exponent written, then the exponent is 1. The exponent on the coefficients is also a 1. It is a good idea to just write the exponent of 1 to avoid mistakes.
Again, there are many methods to simplifying. Just don’t be a rule-breaker and it will work out. Since there are no exponents on the outside of parentheses, reduce the coefficients and make the exponents positive. Then use the appropriate properties. 4 5
It is still a good idea to apply the exponent outside the parentheses to all of the exponents inside. If the coefficients can be reduced (b), do so first to make the operations more manageable.
We’re almost done. We just need to cover addition and subtraction of expressions with negative exponents. Make the definition of a negative exponent to make the exponents positive. Then to combine fractions using the LCD.