To solve this, all you have to do is multiply -8 by -8, or -8 x -8. So, when you have a negative base, it will always be positive. ![]() If we wanted to simplify 3 -2 we would take the reciprocal of 3. The problem you are having is that multiplying a negative number with a negative number is a positive. 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. Hence, negative exponents imply reciprocals. Negative exponent rule: A negative exponent located in the numerator is changed to positive, and is moved to the denominator. When we want to simplify with negative exponents, we take the reciprocal of the base and make the exponent positive. It is important to note a negative exponent does not imply the expression is negative, only the reciprocal of the base. 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: (In other words, theres another rule that also applies: (ab)x ax bx.) Therefore, (ab3)3 a3 (b3)3 a3 b (33. 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. However, the answer is not just ab9 because the a is inside the parentheses and so the exponent of 3 outside the parentheses also applies to the a as well as to the b3. 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 evaluate algebraic expressions with negative exponents. We have shown that the exponential expression an is defined when n is a natural number, 0, or the negative of a natural number. Write each expression using a positive exponent. We can't divide 0 by 0, this is undefined. A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction barfrom numerator to denominator or vice versa. How to Solve Negative Exponents The law of. If we try to raise zero to the power of zero, we will have a problem. Negative exponent rule: To convert a negative exponent to a positive one, write the number into a reciprocal. This rule states that when you have a negative exponent, you can simplify the expression to get the solution by taking the reciprocal of the base raised to the. The negative exponent’s rule can be easily understood with the example discussed in the image below, These rules can be easily understood by the example discussed below, Example: Simplify 3-3 × 1/(4-2) Solution: Using the above rule for solving negative exponents, a-n 1/a n and 1/a (-n) a n. Here are some rules working with negative exponents that you should be familiar with: 1. We can state that any non-zero number raised to the power of zero is 1. There are a few rules for exponents that are only for negative exponents. 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. 3x5 x5 3 x5 x5 3 x5 5 Apply the quotient rule: subtract exponents 3 x0 Apply the zero exponent rule 3. c3 c3 c3 3 Apply the quotient rule: subtract exponents c0 Apply the zero exponent rule 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. We can apply the zero exponent rule and other rules to simplify each expression: 1. ![]() 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. To simplify a power of a power, you multiply the exponents, keeping the base the same.In our last lesson, we learned about the power rules and product rule for exponents. a.) When the terms with the same base are multiplied, the powers are added. Example 3: State true or false with reference to the multiplication of exponents. This leads to another rule for exponents-the Power Rule for Exponents. Solution: According to the rules of multiplying exponents, when the bases are the same, we add the powers. Once the bases are rewritten as their reciprocals. For example 10\cdot10\cdot10 can be written more succinctly as 10^ The negative exponent rule states that the base with a negative exponent must be written as its reciprocal. We use exponential notation to write repeated multiplication. Repeated Image Anatomy of exponential terms
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