The Definition of Negative Exponents

Wheezy, Macaroni, and Pipsqueak are starting to notice a disturbing trend. The iceberg where they love to play keeps breaking in half! It's a repeated-division catastrophe! Fortunately, Professor Penguin is on hand to explain using the definition of negative exponents. Let's take a closer look at how the iceberg is breaking apart. We started with one iceberg. Dividing this iceberg in half gives use two icebergs that are one-half the size of the original. But the icebergs continue to split in half. One-half of the original, divided by two, gives us one-fourth. One-fourth divided by two gives us one-eighth. So after 3 rounds of splitting in half, each mini-iceberg is one-eighth the size of the original. How small would each berg be after 6 rounds of splitting in half? We can answer this question by looking at exponents and division. You should already be familiar with the Law of Quotients of Powers, which tells us how to divide numbers written in exponential notation, when they both have the same base. For example, how would you simplify 2 to the 5th over 2 squared? When dividing numbers with the same base, we subtract the exponents. So this expression can be simplified to 2 cubed. But let's take a look at what happens if we have a larger exponent in our denominator than in our numerator. To simplify 2 to the fifth divided by 2 to the eighth we subtract the exponents and are left with 2 to the negative third power. But what does that mean? If we write our numerator and denominator in their factored forms and then cancel we can see that 1 over 2 to the third power is the same as 2 to the negative third power. In general, this is how we define negative exponents: x' to the negative 'n' equals 1 over 'x' to the 'n'. So while positive exponents represent repeated multiplication, negative exponents represent repeated division. This will come in handy, since we always want to write numbers without a negative exponent when simplifying. So let's go back to our iceberg. Originally, we had one whole iceberg. After one round of dividing in half, one mini-berg is half the size of the original, which we can write as 2 to the negative first power. After two rounds of dividing, that's 1 over 2 squared, or 2 to the negative second power. After 3 rounds it's one over 2 to the third, or 2 to the negative 3. We want to know how small are the mini-bergs after dividing by 2, six times. How could we write that using negative exponents? Well we're dividing by two over and over again, so that number is our base. And we're going to divide six times, so our exponent is negative 6. Using the definition of negative exponents we can write this as a fraction and see that each mini-berg is one sixty-fourth of the original. “Let's look at three quick examples to see how negative exponents apply to what you already know about the Laws of Exponents." Our first example uses the Product of Powers Law. Four to the negative 3rd times four to the negative 5th. We are multiplying numbers with the same base, so what should do with the exponents? Adding the exponents, gives us 4 to the negative eighth power. We don't want to leave this number with a negative exponent, so we use the definition of negative exponents to simplify to the fraction one over 4 to the 8th. Now an example of the quotient of Powers Law. Three to the negative 1st over three to negative the 7th. Here we are dividing numbers with the same base, so what should we do with the exponents? The Quotient of Powers Law tells us to subtract the exponents, so negative 1 minus negative 7 is equal to negative 1 plus 7. Leaving us with 3 to the 6th power. Our third and final example is a number raised to a power, then raised to a power again. What does the Powers of Powers Law tell us to do with the exponents in this case? A power raised to a power means we multiply exponents. Negative 3 times 2 is negative 6. Because we don't want to leave this number with a negative exponent, we rewrite eight to the negative 6th as one over 8 to the 6th power. Let's review. While positive exponents are how we represent repeated multiplication negative exponents represent repeated division. In general, we define a negative exponent as 'x' to the negative 'n' is equal to one over 'x' to the 'n'. And remember, the laws of exponents apply to all numbers written in exponential notation. This includes Product of Powers Quotient of Powers and Powers of Powers. Uh oh, it looks Professor Penguin got a bit carried away with splitting those icebergs in half. I think we're gonna need another iceberg!