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Examples of negative exponent rule11/24/2023 ![]() In other words, a negative exponent tells us how many times the base appears in a Recall the negative exponent rule, which tells us that any power is the reciprocal of the power having the same base but an The given expression involves the multiplication of two powers that have a different base, both with a negative exponent. Let us summarize these findings in a table.Įxample 2: Evaluating Powers with Negative ExponentsĮvaluate the following expression: 6 ⋅ 2 . If we carry out the division by 7 once more, we get So, we start seeing a pattern here: every time we divide by 7, the exponent decreases by 1. The number 7 is of course simply 7, but it is useful to write the exponent here, as you will see in the If we perform the same division again on 7 , we get Let us look at what happens when we divide the power of a number by this number, for instance, 7 divided by 7. To understand the meaning of extending the exponents to negative numbers, So far, we have assumed that the exponent is positive. The question now is whether we could express the result of the previous question, 1 7, as a power of 7. So, here, the eleven 7s in the numerator cancel out withĮleven out of the twelve 7s in the denominator to give 1. Now, we know that one 7 in the numerator cancels out with one 7 in the denominator sinceħ 7 = 1. We have already simplified our expression to Similarly, we find that theĭenominator is 7 . This is the product rule, which states that ![]() The factor 7 and that the factor 7 would appear 3 + 2 + 6 = 1 1 times. By expandingĪll the powers, we see that the numerator would consist of a repeated multiplication of We notice that in this expression, all the powers have the same base: 7. This would give us 1/3.Example 1: Quotient of Powers of the Same Base If we wanted to simplify 3 -2 we would take the reciprocal of 3. Another way to think about this is by stating that we will drag the base and exponent across the fraction bar and make the exponent positive. When we want to simplify with negative exponents, we take the reciprocal of the base and make the exponent positive. This was just to give you an understanding of where our simplified result comes from. Obviously, we will not be going through all this division each time we need to simplify with negative exponents. Divide by the base (3) each time we reduce the exponent by 1: 1 would be divided by 3, and could be written as 1/3:Īs we continue to decrease our exponent by 1, we continue the same process. What happens if we continue and decrease the exponent by 1 to (-1)? We would continue the pattern. ![]() Therefore, we say zero raised to the power of zero is undefined. We can't divide 0 by 0, this is undefined. If we try to raise zero to the power of zero, we will have a problem. We can state that any non-zero number raised to the power of zero is 1. So what happens when we get to 3 0? We continue the same pattern. ![]() If we want 3 1, we can divide 9 by 3 to obtain 3. If we move to 3 2, we can divide 27 by 3 to obtain 9. When we go from 3 4 (81) to 3 3 (27), we could just divide 81 by 3 to obtain 27. This is because we are removing a factor of 3 when we decrease the exponent by 1. What is the value of 3 to the power of (-4)? To understand negative exponents, let's think about a pattern:Įach time we reduce our exponent by 1, we divide by our base of 3. What happens if we see something such as: Negative Exponents & the Power of Zero Up to this point, we have only dealt with whole-number exponents larger than 1. In this lesson, we will expand on our knowledge of the rules of exponents and learn about negative exponents, the power of zero, and the quotient rule for exponents. In our last lesson, we learned about the power rules and product rule for exponents.
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