03 - Negative Exponents & Powers of Zero (Laws of Exponents), Part 1
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- เผยแพร่เมื่อ 25 มี.ค. 2019
- View more at www.MathTutorDVD.com.
In this lesson, you will learn how to simplify expressions that involve negative exponents in algebra. First, we will show what a negative exponent is and how it relates to an exponent of zero (power of zero).
Negative exponents represent small numbers, while positive exponents represent large numbers. We will simplify numerous expressions with zero and negative exponents in preparation for solving equations with a negative exponent.
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This has always been the case. For me maths teachers always just presumed that everyone understood I teach maths now and I often get accused of going too deep but all I’m doing is trying to make sure that the small minority of my students understand
That true
My math teacher was great, but I had really bad foundations. And more than that, I was always asking why. Like a toddler: why, why, why? This video explained the actual reasoning behind the rules and I could not be more grateful! I’m watching the whole series, thank you so much.
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You’re an excellent instructor. Thank you so much! I’m not a student; I’m an adult. Math was always my favorite subject, particularly algebra. Math was the one subject I never had to study outside the classroom. They way I saw it, when we learned a different topic it was a matter of learning a formula and our homework was simply plugging different numbers (in a variety of forms0 into that one formula. It was as easy as breathing air. I loved the way we learned a simple formula and constantly build on it to make more complex formulas. I was never really able to use or understand, shortcuts; I needed to know exactly why something worked, not just how to do it. I’m thrilled that you explained both and that I was able to understand them. I remember in algebra class, I was the only one who knew a number with an exponent of 0 is 1 but I didn’t know why until today. At the time, I had read it somewhere but the book didn’t explain why. Like many people, had my education interrupted by chronic illness. My biggest regret in life is not having the opportunity to advance in my s education to my fullest potential. Ironically enough, I was able to teach myself enough (from library books; the Internet didn’t exist) to win a full scholarship to college but was derailed by illness again again in my third year.
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Excellent elucidation
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This is really helpful! Especially when taking higher math courses like calculus1, 2 and 3.
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widgeonslayer, I. too, am taking college algebra after a long absence from school. In my experience, it's amazing what you can forget in 40 years.
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Thank you for explaining Powers of Zero. It helps to make sense of math. It also builds conceptual learning which is missing in America's School System.
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Fantastic video as usually!. Building on what you said on anything (bar 0) raised to the 0th power being 1, Khan academy did a good video on it, expanding on what that could mean. He said that you could look at the exponent as how many times you need to multiply 1 by that base power. So 2^3 could be seen as 1 needing to be multiplied by 2 3 times, since you need to multiply 1 by 2 3 times to get your answer; that'd be 1 x 2 x 2 x 2, so 8. So for 2^0, the amount of times you need to multiply 1 by 2 to get your answer would be not at all, so the calculation would just be summed to 1, since you'd need to multiply 2 by 1 zero times.
2^0 = 1
2^1 = 1 x 2
2^2 = 1 x 2 x 2
2^3 = 1 x 2 x 2 x 2
I haven't done the explanation justice, so if anyone's still struggling, Google "The 0 & 1st power - Khan Academy".
In my experience, the more steps you take to work something out the more the chance of making a mistake. It's best to learn that any fraction raised to a negative power is the reciprocal raised to the same power but positive. For example (3/5) to the power of -3 is equal to (5/3) to the power of 3. This way, you get rid of double stacks of fractions, working them faster and having fewer steps to make mistakes in.
Not so. As he said, it’s best to know why something works, not just how to do it, if only for instances when you come across an equation where the shortcut doesn’t work or more importantly, you’re given a scenario that requires you to set up the equation yourself based on the information requested. This is the case in many standardized tests and entrance exams for many institutions and companies. In such a case you need to understand the theory. Not just know a method. As he said, there are often many ways to solve a problem. If you understand the theory, you can determine the best way to solve the problem, which may or may not allow a shortcut. It’s best to know why before you learn how and it’s not as if he said not to ever use shortcuts,only to know the theory first. There’s another TH-cam channel, MindYourDecisions, that illustrates this well. He provides mathematical puzzles many of which look deceptively simple. Many require you to set up the equation which means you have to know why, not just how.
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But what if the exponent is a fraction and is a negative multiplied by a negative?
Ex.(a^2/3)^-3
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Rule 1: 09:11
Rule 2: 12:18
Shortcut for fractions with negative powers in the denominator and numerator: 30:41
Thanks!
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You can see that n^0 is 1 even before seeing the proof just by remembering that anything multiplied by 1 is equal to itself. What this rule clues us into is that arithmetic is the science of counting by one, everything is based on increasing or decreasing by increments of one, or fractions of one. All the tricks they teach us, like the reciprocal, cross multiplying, moving negative signs of fractions, are just clever applications of fractional ones. The word "unit" is related to the word "unity".
For example, the number 8 is really 8(1) or (8x1)/1, eight ones. So 8^3 is 8x8x8x1, you can see that if you remove three 8's you're left with one as 8^0, because you got rid of all the eights. 8^1 is eight times bigger than one, then 8^-1 or 1/(1x8) is 8 times smaller, and 8 only exists as a comparison or ratio to 1, as any number is some quantity related to one. You can see that you are just manipulating the unit one, counting with one, then either multiplying it by some number over and over, or dividing it by some number over and over. The negative exponent is an example of extending the meaning of a preexisting symbol, since division is related to subtraction. Edit: Also forgot to mention, negative just mean opposite, so in the case of 8/1 vs 1/8 the opposite of eight times bigger is eight times smaller, therefore 8^1 vs 8^-1.
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