Negative Exponents

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?

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..

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.

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.