State 1start law of exponents
Answers
Step-by-step explanation:
(a+b)^2 = a^2 + b^2 + 2(a+b)
Answer:
First Law of Exponents
And that right there is one of our laws of exponents. Multiplying two powers of the same base means that we can add the exponents. Now the next question concerns what happens when we divide powers. Again, don’t think in terms of just an abstract law, let’s go back and think this through in terms of the fundamental definition of an exponent. If you understand this law from the perspective of–fundamentally, what is an exponent?
Then you will really understand it. So, I’m going to suggest starting with numbers first. Let’s say 12 to the 7th divided by 12 cubed. How would we figure this out? Well, 12 to the 7th has to be seven factors of 12 multiplied together. We have to have that in the numerator.
In the denominator, we’re gonna have three factors of 12 multiplied together. So we’re gonna have something like this. Well obviously, we’re gonna get some cancellation here. We’re gonna be able to cancel, 1, 2, 3 factors of 12 in the numerator and denominator. So everything in red there, all that cancels.
And, of course, when you cancel, it just becomes 1. So it’s 1 and everything else. And so we’re just left with four factors of 12. And that’s 12 to the fourth. So 12 to the 7th divided by 12 to the 3rd equals 12 to the 4th right there. That suggests the rule.
We’ll make this a little more abstract, 12 to the m over 12 to the n. Well here we’re going to have a fraction and in the numerator of the fraction, we have m factors of 12. In the denominator, we have n factors of 12 and we’re going to assume that m is greater than n at least in this video. This means that n factors of 12 will cancel, So all the factors in the denominator will cancel, and that will remove some of the factors from the numerator.
And what will be left in the numerator after we remove those n factors that cancel, we’ll be left with m minus n factors of 12. And so what we’re gonna be left with is just 12 to the m minus n. Now we can do that entirely in variables, think through in variables we have a to the m divided by a to the n. So this is a fraction in the numerator we have m factors of a, in the denominator we have n factors.
For this video, again, we’re going to assume that m is greater than n. So all those factors in the denominator will cancel. When we take the m factors in the numerator and remove the n factors that cancel, we’re going to be left with m minus n. And that’s going to be the exponent of a. And right there is our second law of exponents.
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