Lesson Video: Simplifying Algebraic Expressions: Negative and Fractional Exponents | Nagwa Lesson Video: Simplifying Algebraic Expressions: Negative and Fractional Exponents | Nagwa

Lesson Video: Simplifying Algebraic Expressions: Negative and Fractional Exponents Mathematics

In this video, we will learn how to use the rules of negative and fractional indices to solve algebraic problems.

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Video Transcript

In this video, we will learn how to use the rules of negative and fractional indices to solve algebraic problems. In order to help us understand these rules, let’s begin by recalling the rules for multiplication and division of exponents. Here we have the rules for multiplication and division of exponents. For values of 𝑚 and 𝑛 in the real numbers, then we can multiply exponents with the same base, and we do that by adding their powers. Then, to divide exponents that have the same base, we subtract their powers. This time, 𝑎 has to be a nonzero value. Now, since 𝑚 and 𝑛 can be any real values, then these rules will also apply for negative and fractional indices. Let’s consider what happens when we modify these rules so that we obtain a negative exponent.

Using the division law, let’s take the value of 𝑚 to be equal to zero. Then, we would have 𝑎 to the power of zero divided by 𝑎 to the power 𝑛. Using this exponent rule, we can write this as 𝑎 to the power of zero minus 𝑛. This is equivalent to 𝑎 to the power negative 𝑛. So let’s consider what we have just discovered by using the fact that 𝑎 to the power zero equals one. It’s the fact that one divided by 𝑎 to the power 𝑛 or one over 𝑎 to the power 𝑛 is equal to 𝑎 to the power of negative 𝑛. We can add this to the rules of exponents as a rule for negative indices. Notice that here 𝑚 and 𝑛 are still values in the real numbers and 𝑎 is nonzero.

In the first example, we will see how we can apply this exponent rule.

Which of the following is equal to negative 10 over nine 𝑥 to the power negative two 𝑦 to the power negative seven? Option (A) negative nine over 10𝑥 squared 𝑦 to the power seven. Option (B) negative 10 over nine 𝑥 to the power seven 𝑦 squared. Option (C) negative 10 over nine 𝑥 squared 𝑦 to the power seven. Or option (D) negative 10𝑥 squared 𝑦 to the power seven over nine.

In this question, we have some negative exponents. And so it would be useful to recall the law of exponents for negative exponents. This law states that 𝑎 to the power negative 𝑛 is equal to one over 𝑎 to the power 𝑛 for any 𝑎 which is nonzero. Because both 𝑥 and 𝑦 have negative exponents here, then we can apply the rule to both variables. For 𝑥 to the power negative two, we can substitute 𝑎 is equal to 𝑥 and 𝑛 is equal to two. So 𝑥 to the power of negative two is equal to one over 𝑥 squared. In the same way, for 𝑦 to the power negative seven, that means that 𝑎 is equal to 𝑦 and 𝑛 is equal to seven. So 𝑦 to the power negative seven is equal to one over 𝑦 to the power seven.

We can then substitute these values into the expression. This gives us negative 10 over nine times one over 𝑥 squared times one over 𝑦 to the power seven. And when we’re multiplying fractions, we multiply the numerators and multiply the denominators. The expression is therefore equal to negative 10 over nine 𝑥 squared 𝑦 to the power seven, which was the answer given in option (C).

In the next example, we will use the exponent rule for negative indices as well as the rule of division for indices.

True or False: The simplified form of 𝑥 to the power negative four over 𝑥 to the power negative two is one over 𝑥 squared.

One way in which we can simplify this expression is by using the division rule for exponents. This rule tells us that if we are dividing two exponents, in this case 𝑎 to the power 𝑚 divided by 𝑎 to the power 𝑛, then we subtract those indices such that we get 𝑎 to the power 𝑚 minus 𝑛 for any 𝑎 which is nonzero. And we know that this value of 𝑥 to the power negative four or 𝑥 to the negative fourth power over 𝑥 to the power negative two is equal to 𝑥 to the power negative four divided by 𝑥 to the power negative two. Using the exponent rule with 𝑚 equal to negative four and 𝑛 equal to negative two, we would have 𝑥 to the power of negative four minus negative two. This simplifies to 𝑥 to the power negative two.

This is a perfectly valid equivalence, but it doesn’t match the form that we were given in the question. However, we should remember that there is another law of exponents for negative exponents, which states that 𝑎 to the power of negative 𝑛 is equal to one over 𝑎 to the power 𝑛 for a nonzero value of 𝑎. So this value of 𝑥 to the power of negative two is actually equal to one over 𝑥 squared. That’s using a value of 𝑛 equal to two. So what we have done here is work out that this expression 𝑥 to the power negative four over 𝑥 to the power of negative two is indeed equivalent to one over 𝑥 squared. And so the statement in the question is true.

But there’s also another way in which we could’ve manipulated this expression to come to the same conclusion. In this alternative method, we would actually start by applying this rule for negative indices first. When we do this and we take the numerator of this expression first, we can say that this is equivalent to one over 𝑥 to the power four. The denominator of 𝑥 to the power negative two is equivalent to one over 𝑥 squared. Simplifying this is easier if we remember that a fraction is the same as a division. So this would be equal to one over 𝑥 to the fourth power divided by one over 𝑥 to the second power.

We divide two fractions by multiplying by the reciprocal of the second fraction. So we are left with 𝑥 squared over 𝑥 to the power four. In order to simplify this even further, we could apply the first rule that we saw in this question. With a value of 𝑚 equal to two and 𝑛 equal to four, we would have 𝑥 to the power of two minus four. And of course two minus four is negative two. And then we can rewrite this using our rule for negative indices as one over 𝑥 squared. And thus, we have confirmed that the statement in the question is true.

Before we look at any more examples, let’s recap some more rules of exponents. These exponent rules for powers tell us that for any 𝑚 and 𝑛 in the set of real numbers, we have that 𝑎 to the power 𝑚 to the power 𝑛 is equal to 𝑎 to the power 𝑚𝑛. 𝑎𝑏 to the power 𝑚 is equal to 𝑎 to the power 𝑚 times 𝑏 to the power 𝑚. And thirdly, 𝑎 over 𝑏 to the power 𝑚 is equal to 𝑎 to the power 𝑚 over 𝑏 to the power 𝑚, where 𝑏 is nonzero.

Let’s see how we can apply these exponent rules in the following examples.

Simplify 𝑚 over 𝑛 to the power negative one all to the power negative three times two 𝑚 to the power negative two over 𝑛 to the power negative two all to the power negative three.

In order to simplify this expression, we will need some of the rules of exponents. Because we have fractions raised to a power, then we can use one of the power laws, which tells us that 𝑎 over 𝑏 to the power 𝑚 is equal to 𝑎 to the power 𝑚 over 𝑏 to the power 𝑚, where 𝑏 is nonzero and 𝑚 is in the real numbers. So let’s apply this rule to the first part of the expression. As we have this fraction to the power negative three, then we know that this will be equivalent to the numerator to the power negative three over a denominator to the power negative three. However, to simplify the denominator of 𝑛 to the power negative one to the power negative three, we’ll need another rule of exponents.

The rule that we need is one of the power laws, which tells us that 𝑎 to the power 𝑚 to the power 𝑛 is equal to 𝑎 to the power of 𝑚 times 𝑛. We take the two exponents of negative one and negative three and multiply them. And we know that negative one multiplied by negative three is three. We have now simplified this part of the expression to 𝑚 to the power negative three over 𝑛 to the power three. Let’s see if we can simplify the second part of this expression in the same way.

The first thing we can do is apply this rule for powers of fractions. So the numerator will be equivalent to two 𝑚 to the power negative two to the power negative three. And the denominator will be 𝑛 to the power negative two to the power negative three. In order to simplify the numerator of this fraction, we will need to recall another law of exponents. This power law tells us that 𝑎𝑏 to the power 𝑚 is equal to 𝑎 to the power 𝑚 times 𝑏 to the power 𝑚, where 𝑚 is in the set of real numbers. The numerator of this fraction will therefore simplify to two to the power of negative three times 𝑚 to the power of negative two to the power of negative three. We can also simplify the denominator, remembering that we can use this second power law here to multiply the exponents. And negative two times negative three gives us six, so the denominator will become 𝑛 to the power six.

The next stage in our working will be to simplify this part of the expression, 𝑚 to the power of negative two to the power negative three. Just as before, we can multiply these exponents. So we have 𝑚 to the power of negative two times negative three. And we know that negative two times negative three is six. Now, we could potentially simplify this expression a little further by dealing with the two to the power of negative three. But for now, let’s substitute these values in orange and pink for the parts of the expression. When we multiply these together, we have 𝑚 to the power negative three over 𝑛 to the power three times two to the power negative three 𝑚 to the power six over 𝑛 to the power six.

We know that when we multiply fractions, we multiply the numerators and multiply the denominators. We might then notice that on the numerator, we have two values of the same base of 𝑚. And there is an exponent rule to help us work this out. This rule tells us that 𝑎 to the power 𝑚 times 𝑎 to the power 𝑛 is equal to 𝑎 to the power of 𝑚 plus 𝑛. Of course, the values that we’re using in the question of 𝑚 and 𝑛 are not the same values that we’re using in these exponent rules. And so on the numerator, we will add the two exponents for 𝑚 of negative three and six. So we have two to the power negative three times 𝑚 to the power of negative three plus six on the numerator. On the denominator, we add the exponents three and six of 𝑛. So we have 𝑛 to the power of three plus six.

At this point, we have simplified the variables of 𝑚 and 𝑛 as much as we can. But let’s see if we can do anything to simplify this two to the power of negative three. And we can use one final exponent rule for negative indices. This rule tells us that 𝑎 to the power of negative 𝑛 is equal to one over 𝑎 to the power 𝑛. This means that two to the power of negative three can be written as one over two cubed. We should remember that two cubed is equal to two times two times two, and that’s eight. So two to the power of negative three is equal to one over eight. And when we plug in one-eighth in place of two to the power negative three, we have the expression 𝑚 cubed over eight 𝑛 to the power nine. And this is the answer. We have simplified the given expression as much as we can to give 𝑚 cubed over eight 𝑛 to the power nine.

Let’s now see one final example.

Simplify the expression 𝑥 to the power eight over 𝑦 to the power negative four all to the power one-half.

In order to simplify this expression, we can start with the power law for fractions, which tells us that 𝑎 over 𝑏 to the power 𝑚 is equal to 𝑎 to the power 𝑚 over 𝑏 to the power 𝑚. We can therefore write that this expression is equal to 𝑥 to the power eight to the power one-half over 𝑦 to the power negative four to the power one-half. In order to simplify the powers on the numerator and denominator, we can apply a second exponent rule. We multiply the exponents on the numerator of eight and one-half and the exponents on the denominator of negative four and one-half. This gives us 𝑥 to the power four over 𝑦 to the power negative two. While this is a perfectly valid and fully simplified expression, it is commonplace to give negative exponents instead as a positive exponent.

We recall that 𝑎 to the power of negative 𝑛 is equal to one over 𝑎 to the power of 𝑛, where 𝑎 is nonzero. This means that we can write our expression as 𝑥 to the power of four over one over 𝑦 squared. We can simplify this fraction within a fraction by remembering that fractions are all about division. What we actually have here is the expression 𝑥 to the power of four divided by one over 𝑦 squared. We remember that to perform a division by a fraction, we multiply instead by its reciprocal. That means we have 𝑥 to the power of four multiplied by 𝑦 squared. And so the answer is that the expression given in the question can be simplified to 𝑥 to the power of four 𝑦 squared.

Before we finish this video, there are two more rules of exponents that we need to know. These are regarding fractional indices. The first rule tells us that 𝑎 to the power of one over 𝑛 is equal to the 𝑛th root of 𝑎 for any value of 𝑎 greater than or equal to zero and any positive integer 𝑛. We can also extend this to give us a second rule that 𝑎 to the power of 𝑚 over 𝑛 is equal to the 𝑛th root of 𝑎 to the power 𝑚. And that’s also equivalent to the 𝑛th root of 𝑎 to the power 𝑚. These two rules are particularly useful when we’re using numerical values rather than algebraic values.

But now let’s summarize the key points of this video. We saw that the laws for multiplication, division, and powers of indices apply also for fractional and negative indices. We then saw that the law of exponents for negative indices is one over 𝑎 to the power 𝑛 is equal to 𝑎 to the power of negative 𝑛 for nonzero values of 𝑎. And finally, we finished with two laws of exponents for fractional indices. It is worth noting all of these exponent laws as we are often required to learn them for examinations.

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