Algebra

Zero, negative, and fractional exponents | Pre-Algebra | Khan Academy

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I previously told you that any number raised to the power 0 equals 1 So x ^ 0 = 1 And I gave you a proof that explains this case If we have 3 ^ 1 It is equal to 3 3 ^ 2 = 9 3 ^ 3 = 27 So every time we decrease the power value, we divide by 3 27 ÷ 3 = 9 9 = 3 = 3 Then 3 ÷ 3 = 1 And so we get the result of 3 ^ 0 This is how we think And we will need this method to think How the properties of the foundations work For example, I told you that a ^ b x a ^ c = a ^ b + c Now, what would happen if c = 0 Or what would happen if we had a ^ b x a ^ 0? Well, according to this feature, this will be equal a ^ b + 0, which equals a ^ b So a ^ b x a ^ 0 = a ^ b So if you divide the two sides by x a– let me rewrite This– a ^ b x a ^ 0, if we use this property The result is a ^ b, right? b + 0 = b If you divide the two sides by a ^ b, then what will you get? On the left side, we will have a ^ 0, right? And these are deleted a ^ 0 = 1 And you can use a similar proof that applies to all Fundamentals properties, which have an elevated power setting of 0 And equal to 1 This makes sense when we divide by 3, in each step we perform Decreasing the value of the exponent in it And it will always succeed When we have 3 ^ -1, we’ve seen this in The last symptom is equal to 1/3 ^ 1 Or 1/3 So again, when moving from 3 to 0 to 1/3 We divide by 3 at a time So this really makes some sense as that 3 ^ 0 = 1 But this creates a gap What is the output of 0 ^ 0? This is a strange idea That is, we will multiply 0 by itself 0 times This depends on the context you are using Sometimes people can say that the value is unknown, but others And they are many, at least in my area of \u200b\u200bexpertise, who will say the outcome Equals 1 And the reason – though that’s not Obviously, you can write 0 ^ 0 in Online search engine, and it will give you output 1 Even if it is not self-evident, and a reason The output is defined in this way Some formulas succeed One particular formula, which is the binomial equation As this works when I find the coefficients of the binomial equation, I will not explain it Now, then the bottom line is 0 ^ 0 = 1 This is an interesting thing and you have to think about what Means? Let us now talk a little bit about the other characteristics Then we can use them all when doing a group solution Of the issues, I told you in the last show what It means to raise the number of a negative force a ^ -1, or maybe I have to say a ^ -b = 1 / a ^ b Let’s apply this to a set of real examples 3 ^ -3 = 1/3 ^ 3 It is equal to 1/3 x 3 x 3 Any = 1/27 And if I ask you, what is the result of 1/3 ^ -2? Well, that is equal 1/3 ^ 2 We get rid of the negative and turn it over Visawy 1 / – How much is 1/3 x 1/3? 1/9 And equal to– that’s 1 ÷ 1/9 = 1 x 9, then = 9 And this is very logical, because 1/3, remember, 1/3 Equal to 3 ^ -1, right? 3 ^ -1 = 1/3 ^ 1 Equivalent to 1/3 If we substitute 1/3 for 3 ^ -1, the result is (3 ^ -1) ^ – 2 These two phrases are equal And if we use one of the characteristics that we learned in The first lesson, we will get an output These two heads So it equals 3 ^ -1, x -2 It becomes positive 2, and the result is 9 This is something that is organized as all the characteristics of the foundations Beautifully connected, not Conflict with one another It does not matter which property you use, in the end you will get To the correct answer, as long as you are not No mistake Now, the last thing I want to define is understandable Fractional exponent So if we have a raised number of the breaking force – we assume that We have a ^ 1 / b I will know this This is equal to the root of b for a So let me be more clear here And I’ll use the numbers here So if I say 4 ^ 1/2, that means The square root of 4 This is equal, if intended, of course, the root root Equal to 2 If I say, I would be clear, 8 ^ 1/3 This is equivalent to the cube root of 8 This is, to some degree, one of the things Basically troublesome Here I say, what is the number if you hit him 3 times Will equal 8? If you assume x = 8 ^ 1/3, then this is Equivalent to x ^ 3 = 8 How do I know that these two phrases are equal? Well, I can raise both sides of the equation To the power 3 If you apply that to the left side and raise it to the 3rd force Then on the right side, what will I get? On the left side, I’m going to get x ^ 3 And on the right side, get 8 ^ 1/3 x 3 Ie 3/3, and equal to 1 If x = 8 ^ 1/3, what is the value of x? Well, 2 x 2 x 2 = 8 There is no easy way, especially when dealing with The fourth root, or the fifth, or when we have decimal numbers To calculate it Perhaps most of the time you will need a calculator to help you calculate this But the numbers are like 1/8 3/4, 16 ^ 1/4, or 27 ^ 1/3 It is not difficult to calculate it So this, I’ll be clear, = 2 Now, let’s make things a little trickier How much result 27 ^ -1 / 3? Don’t worry too much I will solve it step by step When we have negative power, this is Equals 1/27 ^ 1/33 These two are equal We get rid of the negative signal and we get 1 / All of this Then what is the outcome of 27 ^ 1/3? Well, how many times if you multiply it by yourself 3 times do you get 27? It’s 3 This is equal to 1/3 Not bad Now I will make it a little more difficult To get a little tedious Let me solve an interesting example How much is the output of 8 ^ 2/3? This looks scary All you have to remember is that this is similar It can be solved using the foundations rules It is the same (8 ^ 2) ^ 1/3 How do I know this? Well, if you hit these two pins So 8 ^ 2/3 equals 8 ^ 2, then We raise the whole amount to the power of 3 But we can imagine it in another way This is equal to (8 ^ 1/3) ^ 2 Because what method we will use, when we hit the foundations We’ll get 8 ^ 2/3 Let’s check that we are We will get the same value So 8 ^ 2 = 64 Then we raise the magnitude to the 1/3 force And we have here, 8 ^ 1/3 We have already created the output Equal to 2, because 2 ^ 3 = 8 So it is 2 ^ 2 Now, how much is 64 ^ 1/3? What number multiplied by himself 3 times and gives you the result 64? Well, 4 x 4 x 4 = 64, or 4 ^ 3 = 64, which means 4 = 64 ^ 1/3 Equals 4 Fortunately for us, 2 ^ 2 also equals 4 So the way we do it is not important Where you can take the square root and then the cubic root, or You can take the third root and then square it And you’ll get the same output Now, all that I have dealt with It was an actual preparation Now let me deal with issues Contains variables So we want to take some phrases and We make sure that The basis for the result is not negative So let me say x + 3 / x ^ -7 This issue can be viewed in several ways We can consider it x ^ -3 x 1 / x ^ -7 How much is the result of 1 / x ^ -7? This is x ^ 7, right? If we have 1 divided by a number, you can get rid of 1 By placing a negative sign against the exponent But if you put a negative sign in return -7, you would get x ^ 7 This can be simplified to x ^ -3 x x ^ 7 Then we can gather the foundations, so we have x ^ 4 Now we’re going to use another method, which is really logical Where we can lay the foundations Well, we have the same basis So we have x ^ -3 – – 7 Well, -3 – -7, equal 3 + 7 = x ^ 4 And the last way, – I mean, there it is More than one method can be followed here We could say x ^ -3 / x ^ -7– Sorry, not -x – / x ^ -7 Well, x ^ -3 equals 1 / x ^ 3 This phrase – x 1 / x ^ -7 So this is equal 1 / x ^ 3 x x ^ -7 You can gather the foundations, so they become 1/3 – 7 = x ^ -4 And then– if we get rid of the inverse, we take Its inverse, and we can put a negative sign against this Minus, so it becomes positive – so it is Output x ^ 4 So it doesn’t matter which way we go, if we are in line with Rules, and we get x ^ 4 Let us now solve a complex issue Then I think we will have accomplished what is required of us today Say: 3x ^ 2 x y ^ 3/2 We’ll divide it by x x y ^ 1/2 Again, this is equivalent to 3 x phrases So 3 x x ^ 2 / x x y ^ 3/2 / y ^ 1/2 Well, this is equal to 3 x – how much x ^ 2 / x? Or x ^ 2 / x ^ 1? This is equal to x ^ 2 – 1 And then this becomes y ^ 3/2 – 1/2 So what is the final outcome? It’s 3 x x 2 – 1 = 1– and I could write x here– x 3/2 – 1/2 = 2/2 So y ^ 2/2 2/2 is the same as y This is equal to 3xy In any case, I advise you to do more Examples of this But you will see that this is just a use of the rules I explained it in the last lessons, so that you can To simplify any exponential expression That’s all I have now, see you soon I have told you in many lessons that any number raised to power 0 equals 1

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