Download Power to Power Rules and Negative Exponents What happens

Power to Power Rules and Negative Exponents
What happens when I raise a power to another power? For example, (32)5 which is read as 3
squared raised to the third power (pretty simple, right). If I wrote this in expanded form it
(3*3)*(3*3)*(3*3)*(3*3)*(3*3)
If you notice, I have 5 groups of 32
If I wrote that problem in exponential form I would have 310 and if I wrote it in standard
form, I would have 59,049.
Look closely at the original problem, (32)5, and the solution, 310. Can you determine a
shortcut that we can use instead of writing it out in expanded form? If you can, write it
down and feel free to share with your peers.
Let’s try a few practice problems:
a. (75)3 =
b. (x5)3 =
c. (x-3)8 =
d. (x5)10=
Seems pretty simple but what happens if I have a coefficient other than 1? That a great
question, you’re just full of them this week. We have to remember that any whole number can
be written in exponential form by simply using 1 as the exponent. For example, 5 is the same
thing as 51. Let’s apply this concept to the problem below.
(5x3)4 can be written as (51x3)4. Using the concepts from above, I can simply
multiply all the exponents so I would end up doing (51)4 and (x3)4 and end up with 54 or 625 and
x12 for a final answer of 625x12. If you notice, the outside exponent, 4, actually got multiplied by
both of the interior exponents, 1 and 3.
Try these out and see if the concept still makes sense.
a. (3x11)3=
b. (-2x5)4=
c. (8x-3)5=
Little trickier now because this combines today’s and yesterdays concepts:
d. (5x2)7 * (2x4)6 =
Negative Exponents
How can an exponent be negative? Look what happens when you divide a larger number by a
smaller number.
43 = 43-5 = 4-2
45
So, what exactly does a negative power mean? Let’s look at the in-between steps in the
example above, plus a few more examples.
1
1
1
1
43 = 4 ∙ 4 ∙ 4
= 43-5 = 4-2 = (4)2 = 4 ∗ 4 = 16 = .0625
45
4∙4∙4∙4∙4
1
1
1
1
1
95 = 9 ∙ 9 ∙ 9 ∙ 9 ∙ 9_______ = 95-8 = 9-3 = (9)3 = 9 ∗ 9 *9= 729 ≈ .00137
98
9 ∙9 ∙ 9 ∙9 ∙ 9 ∙ 9 ∙9 ∙ 9
6x4 = 2 ∙ 2 ∙ 2 ∙ 2_______________ =
2x10 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2
6 4-10
x
3
1
2
= 2x-6 = 2(𝑥)6 = 𝑥 6
How are negative exponents different from negative numbers? What can you say about the
value of exponential terms with negative numbers?
How can you convert a negative exponent to a positive one, while keeping the same value of
the number?
Example:
5-2 =
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What if the negative exponent was on a number in the denominator (bottom) of a fraction?
How can you convert it to a number with a positive exponent?
Example:
1 =
5-2
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Practice: simplify each expression using only positive exponents.
1.
12𝑥 11
3𝑥 19
=
2. (3x-3)4 =
3. 4x-3y7 =