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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 = _______________________________________________________________________ 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 _______________________________________________________________________ Practice: simplify each expression using only positive exponents. 1. 12𝑥 11 3𝑥 19 = 2. (3x-3)4 = 3. 4x-3y7 =