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

In order to help you understand the rules of negative and fractional indices we will first recall the rules for the multiplication and division of exponents.

### Rules of Exponents: Multiplication and Division

The rules for the multiplication and division of exponents are as follows.

- Multiplication of exponents with the same base: , where and can take any value in the real domain.
- Division of exponents with the same base: , where and where and can take any value in the real domain.

Since and can take any value in the real domain, these rules apply for negative and fractional indices. We will first consider what happens when we modify these rules such that we obtain a negative exponent.

Using the division law for exponents, where , we can see that when , this will result in a negative index.

Further, if we let , we can see

Recall that when , so we get the following formula:

Or, written as a fraction, we get the following:

This then leads us to our next rule of exponents for negative indices.

### Rules of Exponents: Negative Indices

The rule of exponents for negative indices is as follows: where and can take any value in the real domain.

When writing algebraic expressions with exponents, it is convention to simplify answers in such a way that the exponent is positive. In the case of , we would rewrite this as when giving a final answer. Note that this may not always be the case, but it is useful to consider.

In the following example, we will apply the rule of exponents for negative indices.

### Example 1: Rewriting Algebraic Expressions Using Laws of Exponents with Negative Exponents

Which of the following is equal to ?

### Answer

In order to rewrite the expression we must use the law of exponents for negative exponents, which states that

As both and have negative exponents, then we will apply the rule to both variables.

For , we can substitute and , which gives us

For , we can substitute and , which gives us

Now we have rewritten both variables with positive exponents, we can then put them back into the original expression, giving us

Hence, the answer is option C, .

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

### Example 2: Matching Two Expression Using Laws of Exponents with Negative Exponents

True or False: The simplified form of is .

### Answer

In order to simplify , we start by using the division rule for exponents:

In this case, even though both exponents are negative, with and , we can still apply this rule. It also helps to rewrite the fraction as a division here. By doing so we get

Next, since the index is still negative, then we use the law of exponents for negative exponents, which states that

By substituting and into the law, we get

Therefore, , so the answer is true.

Looking back at the previous example, there are other ways of simplifying to get .

Another approach is to use the law of exponents for negative indices first, which states that

We then substitute and for the term in the numerator and and for the term in the denominator. This gives us

We can then use our understanding of fractions to write this as

We then use the rule for division of exponents, which states that

With , , and , we get

This shows that we can apply the rules in different order and obtain the same equivalent expression.

In the next example, we will simplify an algebraic expression using the laws of exponents, including the power law, the division law, and the law for negative indices. Let’s recap the power laws first.

### Rules of Exponents: Further Power Laws

The rules of exponents for powers of exponents are

- ,
- ,
- , ,

where and can take any value in the real domain.

### Example 3: Simplifying Rational Algebraic Expressions Using Laws of Exponents with Negative Exponents

Simplify .

### Answer

In order to simplify this algebraic expression, we consider which laws are required. As there are fractions raised to a power, then we can first use the power law that states that where and can take any real value.

When applying this law to the first part of the expression, , we get

We can simplify the denominator further using the law of exponents for powers, that states: where and can take any real value.

So, we get

Similarly, we can apply the law for powers of fractions to the second part of the expression, , which gives us.

As with the first part of the expression, we can simplify the denominator using the law of exponents for powers, but for the numerator, we use the law for powers of products, which states that where can take any real value.

So, we get

When combining the two parts of the expression, we get

Next, we use the multiplication rule for exponents, which states that where , , and are any real value.

We can then simplify the numerator and denominator, as some of the components have the same base:

Lastly, we use the rule for negative exponents to simplify . This rule states that where and can be any real number.

So, we then have

Therefore, .

So far, we have met negative indices. Next, we will consider fractional indices.

Let’s consider the expression .

We know that and similarly

Let’s suppose there is an index such that

By the power law for exponents, we know that

We also know that

Therefore,

So, by equating the indices, we get

So,

Since , then we can deduce that

Therefore, we can see that any number raised to the power of a half is equal to the square root of . In other words,

We can then follow the same steps for the general root in the equation .

We know that for any positive integer : and similarly

Let’s suppose there is an index such that

By the power law for exponents we know that

We also know that

Therefore,

So, by equating the indices, we get

So,

Since , then we can deduce that

Therefore, we can see that any number raised to the power of is equal to the root of . In other words, for any positive integer and .

Having deduced the general root, we can next use the law of exponents for powers to find a law for .

We know the law of exponents for powers is where , and and can take any value in the real domain.

So, using , we can write where , is a positive integer, and is a real value.

The laws we have derived are summarized in the next definition.

### Definition: Law of Exponents for Fractional Indices

The rule of exponents for fractional indices is as follows:

- , for any value of and any positive integer ,
- , for any value of and any positive integer .

Note, that in the laws above that, can be negative, but this is beyond the scope of this explainer as it addresses properties of complex numbers. Further, different results are obtained for negative values of depending on the order of operations of the roots and powers if the index is not simplified. Therefore, it is recommended to simplify fully a fractional index before evaluating.

In the next example, we will consider simplifying with both fractional and negative indices.

### Example 4: Simplifying Expression with Fractional and Negative Indices

Simplify the expression .

### Answer

In order to simplify, we start with the power law for fractions, which states that where , and and can take any real value.

So, we have

So, we then get

Next, we use the law of exponents for negative indices, which states that

This gives us

Therefore, .

In the next example, we will simplify expressions with negative fractional indices.

### Example 5: Simplifying Expressions with Negative Fractional Indices

Simplify the expression .

### Answer

In order to simplify we start with the power law for fractions, which states that where , and and can take any real value.

So, we have

Next, we apply the power law for products to the numerator and denominator, which states that where , , and can take any real value.

For the numerator, this gives us

And for the denominator, we have

Substituting back into the original expression, we get

Next, we use the division rule for exponents, which states that

For ease of simplifying and to avoid error, we will rewrite the expression so like terms are grouped together.

So,

Simplifying the part of the expression with the -terms gives us

Similarly, simplifying the part of the expression with the -terms gives us

Finally, substituting both parts back into the expression, we get

Therefore, .

In this explainer, we have learned about negative and fractional indices, and how to apply the various laws of exponents in order to simplify algebraic expressions.

### Key Points

- The laws for the multiplication, division, and powers of indices apply
also for fractional and negative indices, which are
- ,
- ,
- ,
- ,
- , .

- The law of exponents for negative indices is
- The laws of exponents for fractional indices are
- , for any value of and any positive integer ,
- , for and any positive integer .