Evaluate 0.03125 to the power of negative 0.2.
So to approach this question, we’re going to do three things. We’re going to change our decimal number into a fraction. We might not always need to do this, but it will be helpful in this question. We’re going to change our negative exponent into a positive exponent. And we’re going to change our 0.2 in the exponent into a fraction. We can do these three things in any order. Let’s start by looking at our decimal number. Now, you may say this as the value 0.03125 or as three thousand one hundred twenty-five hundred thousandths. In either case, it’s equivalent to the fraction 3125 over 100000.
And the next step is to simplify it. So let’s divide our numerator and denominator by 25 which is 125 over 4000. We can see since our 125 ends in a five and the 4000 ends in zero, then we must be able to divide by at least five. And in fact, we could see that 25 will go into both numbers. In fact, we can divide again by 25 on our numerator on a denominator giving us five over 800. And the final division by five will give us the fully simplified fraction one over 32. Now, we can write our decimal number to the power of negative 0.2 as one over 32 to the power of negative 0.2. So we have changed our decimal number to a fraction. So let’s change this negative exponent into a positive one.
To do this, we’re going to use the exponent rule that says that if we have a negative exponent, for example, 𝑎 to the power of negative 𝑛, then we can write this as one over 𝑎 to the 𝑛. In other words, we take the reciprocal of a number and the negative exponents becomes a positive exponent. So for our value then, our one over 32 will become 32 and the negative exponent of negative 0.2 will become 0.2. Now, we’ve changed our negative exponent to a positive exponent. Let’s move on then to changing our exponent into a fraction. Thinking about 0.2 then or saying it as two-tenth would give us a fraction two over 10. This could be simplified to one-fifth. Therefore, 32 to the part of 0.2 is equivalent to 32 to the power of one-fifth.
And what do we mean by to the power of one-fifth? Well, that’s the same as taking the fifth root, which we can write as the fifth root of 32. We write this as a small five next to a root symbol. And we’re careful not to confuse it with five times the square root. To find the fifth root of 32, we’re really asking which value of 𝑥 will give us 𝑥 times 𝑥 times 𝑥 times 𝑥 times 𝑥 to give us 32. And since the value two would work, then our answer to the fifth root of 32 is two. So we completed all three steps. And a reminder that we can do those in any order to give us our final answer of two.