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Solving with positive exponents makes sense. But how does solving with negative exponents work? After all, 5^2 is just 5 X 5 – or two 5s multiplied by each other. But how can you multiply negative three 5s by each other? What could the phrase “negative three 5s" mean?
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The Common Mistake
Many students, when confronted with calculating 5^-2, make a simple mistake. They figure that since 5^2 = 25, then 5^-2 must be -25. Watch out for this mistake! Adding a negative sign to the power is not the same thing as multiplying the answer by -1. There’s one basic rule that can help you understand how to calculate negative exponents..
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The Basic Rule
There’s one rule that you have to memorize about negative exponents, and it involves reciprocals. Here’s the rule: Raising a number to a negative power is the same thing as raising the reciprocal of that number to a positive power.
What does that mean? It means that if you were trying to raise 5 to the power of -2, you would first find the reciprocal of 5 – which is 1/5. So when 5 is raised to the power of -2, it is the same thing as saying that 1/5 is raised to the power of positive 2, or (1/5)^2. If you would multiply that out, you would get 1/25. Therefore, 5^-2 = 1/25. Once you remember the negative exponent rule, everything else falls into place.
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Fractions and Negative Exponents
When the base of an exponent is a fraction, you can follow the same logic. For example, let’s say you want to raise 3/4 to the power of -3. To calculate this, you would first take the reciprocal of 3/4, which is 4/3. Then you would raise 4/3 to the power of +3, or (4/3)^3. If you would multiply that out, you would get 64/9.