Video: Fractional Exponents

In this video, we will learn how to simplify fractional exponents and convert between exponential and radical expressions.

14:09

Video Transcript

In this video, we’ll learn how to simplify fractional exponents and try to evaluate fractions and decimals with these fractional exponents. So let’s remind ourselves of what it means to have a number written with an exponent. Here we have three times three, which is written as three squared. The three in the three squared is called the base, and the two is the exponent or the power or the index of the number.

If we look at the number four to the third power, that means that we write the number four three times and multiply them together. When working with exponents, we will need to use the addition rule. For example, if we have three to the fourth power multiplied by three to the fifth power, then our three to the fourth power is written as four threes multiplied together. And our three to the fifth power is written as five threes multiplied together.

So when we multiply these two values together, we have the number three written nine times. So our answer to these multiplied would be three to the power of nine. This can be summarized by the rule 𝑥 to the power of 𝑎 multiplied 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏.

In our example, we had three to the power of four times three to the power of five. And when we added those exponents, we got nine. So we had the answer three to the power of nine.

So what happens when we have a problem such as nine to the power of one-half times nine to the power of one-half? Well, using our addition rule for exponents, we could write this as nine to the power of one-half plus one-half, which is equal to nine to the power of one. And that, in turn, is simply equal to nine. This means that our number nine to the power of a half multiplied by itself must be equal to nine.

A value that could do that would be three multiplied by three, or in other words the square root of nine. Therefore, nine to the power of one-half is equivalent to the square root of nine.

Let’s look at another example, taking 27 to the power of one-third times 27 to the power of a third times 27 to the power of a third. We can then use the addition rule for exponents, which tells us that we add the exponents, giving us 27 to the power of one-third plus one-third plus one-third. Which is equal to 27 to the power of one, which is, in turn, equal to 27. Therefore, the number which multiplies by itself and then by itself again to give 27 must be the cube root of 27, which would be three, since three times three times three is 27.

So we can see that another way of writing 27 to the power of one-third would be the cube root of 27. In fact, the general rule is that if we have a value 𝑥 written to the power of one over 𝑎, it is equivalent to the 𝑎th root of 𝑥. So let’s now use this general rule to evaluate a number written with a fractional exponent.

Evaluate 64 to the power one-third.

To answer this question, let’s use the rule that if we have a value 𝑥 to the power of one over 𝑎, this is equivalent to the 𝑎th root of 𝑥. So we can write our value 64 to the power of one-third as the cube root of 64. We can verify this because we could write 64 to the power of one-third times 64 to the power of one-third times 64 to the power of one-third as 64 to the power of one. We can do this because we’re using the exponent rule, which says that when we’re multiplying values written in exponent form, we add the indices. In this case, our exponents or indices, a third plus a third plus a third, would give us one.

And therefore, this value, which we can multiply by it itself and then by itself again to give us 64, must be the cube root of 64. So what is the cube root of 64? Well, it’s four, since four times four times four is 64. So 64 to the power of one-third is four.

Let’s now have a look at a number written with a fractional exponent where the numerator is not simply one. And that makes it a little bit more interesting. To do this, we’re going to use the rule that we’ve seen before that, to multiply numbers written in exponents form, we add the exponents. Here we can use this formula, but almost as though it’s in reverse. We’re starting with our 27 to the power of two-thirds and splitting it into two numbers, 27 to the power of one-third and 27 to the power of one-third.

In fact, if we wanted to be really mathematically precise, we could write these both with the exponent 𝑎 since it’s the same value. And now we know that we can write 27 to the power of one-third as the cube root of 27. So we have the cube root of 27 multiplied by the cube root of 27. And that’s the same as the cube root of 27 squared.

And have you worked out how it relates to the original question? Well, if we take a value 𝑥 to the power of 𝑎 over 𝑏, it’s equivalent to the 𝑏th root of 𝑥 all to the power of 𝑎. And it’s also equivalent to the 𝑏th root of 𝑥 to the power of 𝑎. If you can’t quite see the difference, in the first case, we find the 𝑏th root of 𝑥 first and then write it to the power of 𝑎. But in the second scenario, we would find 𝑥 to the power of 𝑎 and then take the 𝑏th root of that.

Since these are equivalent, it doesn’t matter which form we use. But very often this first form is simpler because, in this case, we find the root first, the 𝑏th root, which gives us a smaller number, which we then have to find the 𝑎th power of. Looking at our question then, we could’ve said that the cube root of 27 squared is equivalent to finding the square of 27 and then taking the cube root.

In the first form, we’d find the cube root of 27 first, which is three, and then we would square it, which gives us an answer of nine. In the second form, we’d need to square 27 first, which is 729, and then we need to take the cube root of this, which would be nine. So in both forms, we got the same value of nine. But in the first form where we found the cube root first, there was a lot less working out to be done.

So let’s have a look at a question where we use this rule for fractional exponents. And of course, you may want to pause the screen after you’ve seen the question to have a go at it first.

Evaluate 16 to the power three-quarters.

Here we have a number written with a fractional exponent of three-quarters. We can use the rule that if we have 𝑥 to the power of 𝑎 over 𝑏, this is equivalent to the 𝑏th root of 𝑥 to the power of 𝑎. So starting with our base 16, we’re going to take the fourth root. And then we’re going to take that all to the third power. Starting with our fourth root then, we can say that the fourth root of 16 is two, since two times two times two times two is 16. And we then need to take the third power of that. So two times two times two gives us eight. So 16 to the power of three-quarters is equal to eight.

Evaluate 3125 to the power three-fifths.

To answer this question, we’re going to use the rule that 𝑥 to the power of 𝑎 over 𝑏 is equivalent to the 𝑏th root of 𝑥 to the power of 𝑎. So in a fractional exponent, the top number is the power. So here we’ll be taking the third power. And the denominator is the root, so here we’ll be finding the fifth root.

So we start by taking the fifth root of 3125, and then we find the third power of that. Starting with the fifth root of 3125, that’s equal to five, because five times five times five times five times five is 3125. And we’ll then take the third power of five, which means that we’re working out five times five times five. And five times five is 25, times five will give us 125. So 3125 to the power of three-fifths is 125.

In the next example, we’re going to look at a fraction written with a fractional exponent.

Evaluate 125 over 343 to the power of two-thirds.

The first thing to note here is that our fraction, two-thirds, is an exponent or a power. And it doesn’t mean that we’re multiplying it with the fraction 125 over 343. To start, let’s take the exponent of this fraction and write it as an exponent of the numerator and an exponent of the denominator. In other words, we can use the rule that if we have a fraction 𝑥 over 𝑦 to the power of 𝑎, it’s equivalent to 𝑥 to the power of 𝑎 over 𝑦 to the power of 𝑎. So for our value, we can write our numerator as 125 to the power of two-thirds and our denominator as 343 to the power of two-thirds.

So now let’s simplify these fractional exponents of two-thirds. Recall that if we have a value 𝑥 to the power of 𝑎 over 𝑏, this is equivalent to the 𝑏th root of 𝑥 to the power of 𝑎. And therefore, on our numerator, 125 to the power of two-thirds is equivalent to the cube root of 125 squared. Our denominator is equivalent to the cube root of 343 squared.

We can notice on our numerator that this is equivalent to squaring 125 first and taking the cube root. Equally, on our denominator, we could square 343 first and then take the cube root. However, in the second form written in orange, this will have much larger numbers. Since we’re squaring 125 first and then trying to find the cube root of that. Whereas if we start by taking the cube root first and then squaring it, our values won’t get so large.

Therefore, the cube root of 125 will give us five. And since we then need to square it, we’ll have five squared on our numerator. And on our denominator, the cube root of 343 is seven, since seven times seven times seven gives us 343. And then we’ll need to square that. Evaluating our squares then will give us the final answer of 25 over 49.

In our final example, we’re going to see a decimal written with a decimal exponent. To solve it, we’re going to change both decimals into fractions.

Evaluate 0.0625 to the power 0.25.

Our approach to evaluating this will involve taking both of our decimals, the base and the exponent, and writing those as fractions. So 0.0625 is equivalent to 625 over 10000, and our exponent of 0.25 is equivalent to one-quarter. We can then apply some exponent rules.

The first rule we’re going to use is that if we have a fraction 𝑥 over 𝑦 to the power of 𝑎, this is equivalent to 𝑥 to the power of 𝑎 over 𝑦 to the power of 𝑎. So our fraction is equivalent to 625 to the power of one-quarter over 10000 to the power of one-quarter.

Now let’s think about what it means to be to the power of one-quarter. We can use our second rule to help us here, which says that if we have a value 𝑥 to the power of one over 𝑎, it’s equivalent to the 𝑎th root of 𝑥. So our fractional exponent of one-quarter is equivalent to the fourth root.

On the numerator then, we have the fourth root of 625. And on the denominator, it’s the fourth root of 10000. Evaluating the fourth root of 625 gives us five, since if we write five down four times and multiply, we’ll get 625. And then the fourth root of 10000 is 10 since again if we write down 10 four times and multiply, we get 10000. We can then simplify our fraction five-tenths, giving us a final answer of a half.

Let’s now summarize some of the things we’ve learnt in this video. Firstly, we revised how numbers can be written in exponent form, for example, three squared. In this case, three would be the base, and two would be the exponent or power or index. We also recalled and used the rule for multiplying numbers written in exponent form, which is 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏.

We discovered the rule 𝑥 to the power of one over 𝑎 is equal to the 𝑎th root of 𝑥. For example, we saw how nine to the power of one-half is equal to the square root of nine and how 27 to the power of one-third is equal to the cube root of 27. We then discovered a more wide-ranging rule that 𝑥 to the power of 𝑎 over 𝑏 is equal to the 𝑏th root of 𝑥 to the power of 𝑎. We used this rule to help us answer the problem 16 to the power of three-quarters to find that that’s equal to eight.

We then used the rule that if we have 𝑥 over 𝑦 to the power of 𝑎 over 𝑏, we can write this as 𝑥 to the power 𝑎 over 𝑏 over 𝑦 to the power of 𝑎 over 𝑏. We also saw that a number written with a decimal base and/or a decimal exponent can be evaluated by changing the decimals to fractions. We saw this in our last problem where we changed 0.0625 to the power of 0.25 into 625 over 10000 to the power of a quarter. And we were then able to continue solving it.

So now we’ve seen how to evaluate a number written with a fractional exponent. It’s time to go and try some questions for yourself.

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