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

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: ๐‘Žร—๐‘Ž=๐‘Ž๏‰๏Š๏‰๏Šฐ๏Š, where ๐‘š and ๐‘› can take any value in the real domain.
  • Division of exponents with the same base: ๐‘Žรท๐‘Ž=๐‘Ž๏‰๏Š๏‰๏Šฑ๏Š, where ๐‘Žโ‰ 0 and where ๐‘š and ๐‘› can take any value in the real domain.

Since ๐‘š and ๐‘› 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, ๐‘Žรท๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฑ๏Š where ๐‘Žโ‰ 0, we can see that when ๐‘š<๐‘›, this will result in a negative index.

Further, if we let ๐‘š=0, we can see ๐‘Žรท๐‘Ž=๐‘Ž,๐‘Žโ‰ 0.๏Šฆ๏Š๏Šฆ๏Šฑ๏Šwhere

Recall that ๐‘Ž=1๏Šฆ when ๐‘Žโ‰ 0, so we get the following formula: 1รท๐‘Ž=๐‘Ž,๐‘Žโ‰ 0.๏Š๏Šฑ๏Šwhere

Or, written as a fraction, we get the following: 1๐‘Ž=๐‘Ž,๐‘Žโ‰ 0.๏Š๏Šฑ๏Šwhere

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: ๐‘Ž=1๐‘Ž,๏Šฑ๏Š๏Š where ๐‘Žโ‰ 0 and ๐‘› 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 ๐‘ฅ๏Šฑ๏Šจ, we would rewrite this as 1๐‘ฅ๏Šจ 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 โˆ’109๐‘ฅ๐‘ฆ๏Šฑ๏Šจ๏Šฑ๏Šญ?

  1. โˆ’910๐‘ฅ๐‘ฆ๏Šจ๏Šญ
  2. โˆ’109๐‘ฅ๐‘ฆ๏Šญ๏Šจ
  3. โˆ’109๐‘ฅ๐‘ฆ๏Šจ๏Šญ
  4. โˆ’10๐‘ฅ๐‘ฆ9๏Šจ๏Šญ

Answer

In order to rewrite the expression โˆ’109๐‘ฅ๐‘ฆ,๏Šฑ๏Šจ๏Šฑ๏Šญ we must use the law of exponents for negative exponents, which states that ๐‘Ž=1๐‘Ž,๐‘Žโ‰ 0.๏Šฑ๏Š๏Šwhere

As both ๐‘ฅ and ๐‘ฆ have negative exponents, then we will apply the rule to both variables.

For ๐‘ฅ๏Šฑ๏Šจ, we can substitute ๐‘›=2 and ๐‘Ž=๐‘ฅ, which gives us ๐‘ฅ=1๐‘ฅ.๏Šฑ๏Šจ๏Šจ

For ๐‘ฆ๏Šฑ๏Šญ, we can substitute ๐‘›=7 and ๐‘Ž=๐‘ฆ, which gives us ๐‘ฆ=1๐‘ฆ.๏Šฑ๏Šญ๏Šญ

Now we have rewritten both variables with positive exponents, we can then put them back into the original expression, giving us โˆ’109๐‘ฅ๐‘ฆ=โˆ’109ร—1๐‘ฅร—1๐‘ฆ=โˆ’10ร—1ร—19ร—๐‘ฅร—๐‘ฆ=โˆ’109๐‘ฅ๐‘ฆ.๏Šฑ๏Šจ๏Šฑ๏Šญ๏Šจ๏Šญ๏Šจ๏Šญ๏Šจ๏Šญ

Hence, the answer is option C, โˆ’109๐‘ฅ๐‘ฆ๏Šจ๏Šญ.

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 ๐‘ฅ๐‘ฅ๏Šฑ๏Šช๏Šฑ๏Šจ is 1๐‘ฅ๏Šจ.

Answer

In order to simplify ๐‘ฅ๐‘ฅ๏Šฑ๏Šช๏Šฑ๏Šจ, we start by using the division rule for exponents: ๐‘Žรท๐‘Ž=๐‘Ž,๐‘Žโ‰ 0.๏‰๏Š๏‰๏Šฑ๏Šwhere

In this case, even though both exponents are negative, with ๐‘š=โˆ’4 and ๐‘›=โˆ’2, we can still apply this rule. It also helps to rewrite the fraction as a division here. By doing so we get ๐‘ฅ๐‘ฅ=๐‘ฅรท๐‘ฅ=๐‘ฅ=๐‘ฅ.๏Šฑ๏Šช๏Šฑ๏Šจ๏Šฑ๏Šช๏Šฑ๏Šจ๏Šฑ๏Šช๏Šฑ(๏Šฑ๏Šจ)๏Šฑ๏Šจ

Next, since the index is still negative, then we use the law of exponents for negative exponents, which states that ๐‘Ž=1๐‘Ž,๐‘Žโ‰ 0.๏Šฑ๏Š๏Šwhere

By substituting ๐‘›=2 and ๐‘Ž=๐‘ฅ into the law, we get ๐‘ฅ=1๐‘ฅ.๏Šฑ๏Šจ๏Šจ

Therefore, ๐‘ฅ๐‘ฅ=1๐‘ฅ๏Šฑ๏Šช๏Šฑ๏Šจ๏Šจ, so the answer is true.

Looking back at the previous example, there are other ways of simplifying ๐‘ฅ๐‘ฅ๏Šฑ๏Šช๏Šฑ๏Šจ to get 1๐‘ฅ๏Šจ.

Another approach is to use the law of exponents for negative indices first, which states that ๐‘Ž=1๐‘Ž,๐‘Žโ‰ 0.๏Šฑ๏Š๏Šwhere

We then substitute ๐‘›=4 and ๐‘Ž=๐‘ฅ for the term in the numerator and ๐‘›=2 and ๐‘Ž=๐‘ฅ for the term in the denominator. This gives us ๐‘ฅ๐‘ฅ=.๏Šฑ๏Šช๏Šฑ๏Šจ๏Šง๏—๏Šง๏—๏Žฃ๏Žก

We can then use our understanding of fractions to write this as ๏Šง๏—๏Šง๏—๏Šช๏Šจ๏Šช๏Šจ๏Šจ๏Šช๏Žฃ๏Žก=1๐‘ฅรท1๐‘ฅ=1๐‘ฅร—๐‘ฅ=๐‘ฅ๐‘ฅ.

We then use the rule for division of exponents, which states that ๐‘Žรท๐‘Ž=๐‘Ž,๐‘Žโ‰ 0.๏‰๏Š๏‰๏Šฑ๏Šwhere

With ๐‘Ž=๐‘ฅ, ๐‘š=2, and ๐‘›=4, we get ๐‘ฅ๐‘ฅ=๐‘ฅรท๐‘ฅ=๐‘ฅ=๐‘ฅ=1๐‘ฅ.๏Šจ๏Šช๏Šจ๏Šช๏Šจ๏Šฑ๏Šช๏Šฑ๏Šจ๏Šจ

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. Letโ€™s recap the power laws first.

Rules of Exponents: Further Power Laws

The rules of exponents for powers of exponents are

  • (๐‘Ž)=๐‘Ž๏‰๏Š๏‰๏Š,
  • (๐‘Ž๐‘)=๐‘Ž๐‘๏‰๏‰๏‰,
  • ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘๏‰๏‰๏‰, ๐‘โ‰ 0,

where ๐‘š and ๐‘› can take any value in the real domain.

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

Simplify ๏€ผ๐‘š๐‘›๏ˆ๏€พ2๐‘š๐‘›๏Š๏Šฑ๏Šง๏Šฑ๏Šฉ๏Šฑ๏Šจ๏Šฑ๏Šจ๏Šฑ๏Šฉ.

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 ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏‰๏‰๏‰ where ๐‘โ‰ 0 and ๐‘š can take any real value.

When applying this law to the first part of the expression, ๏€ผ๐‘š๐‘›๏ˆ๏Šฑ๏Šง๏Šฑ๏Šฉ, we get ๏€ผ๐‘š๐‘›๏ˆ=๐‘š(๐‘›).๏Šฑ๏Šง๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฑ๏Šง๏Šฑ๏Šฉ

We can simplify the denominator further using the law of exponents for powers, that states: (๐‘Ž)=๐‘Ž,๏‰๏Š๏‰๏Š where ๐‘š and ๐‘› can take any real value.

So, we get ๐‘š(๐‘›)=๐‘š๐‘›=๐‘š๐‘›.๏Šฑ๏Šฉ๏Šฑ๏Šง๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฑ๏Šงร—๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฉ

Similarly, we can apply the law for powers of fractions to the second part of the expression, ๏€พ2๐‘š๐‘›๏Š๏Šฑ๏Šจ๏Šฑ๏Šจ๏Šฑ๏Šฉ, which gives us. ๏€พ2๐‘š๐‘›๏Š=๏€น2๐‘š๏…(๐‘›).๏Šฑ๏Šจ๏Šฑ๏Šจ๏Šฑ๏Šฉ๏Šฑ๏Šจ๏Šฑ๏Šฉ๏Šฑ๏Šจ๏Šฑ๏Šฉ

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 (๐‘Ž๐‘)=๐‘Ž๐‘,๏‰๏‰๏‰ where ๐‘š can take any real value.

So, we get ๏€น2๐‘š๏…(๐‘›)=2ร—๏€น๐‘š๏…๐‘›=2ร—๐‘š๐‘›=2ร—๐‘š๐‘›.๏Šฑ๏Šจ๏Šฑ๏Šฉ๏Šฑ๏Šจ๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฑ๏Šจ๏Šฑ๏Šฉ๏Šฑ๏Šจร—๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฑ๏Šจร—๏Šฑ๏Šฉ๏Šฑ๏Šจร—๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฌ๏Šฌ

When combining the two parts of the expression, ๏€ผ๐‘š๐‘›๏ˆ=๐‘š๐‘›๏€พ2๐‘š๐‘›๏Š=2ร—๐‘š๐‘›,๏Šฑ๏Šง๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฉ๏Šฑ๏Šจ๏Šฑ๏Šจ๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฌ๏Šฌand we get ๏€ผ๐‘š๐‘›๏ˆ๏€พ2๐‘š๐‘›๏Š=๐‘š๐‘›ร—2๐‘š๐‘›=๐‘šร—2๐‘š๐‘›ร—๐‘›.๏Šฑ๏Šง๏Šฑ๏Šฉ๏Šฑ๏Šจ๏Šฑ๏Šจ๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฉ๏Šฑ๏Šฉ๏Šฌ๏Šฌ๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฌ๏Šฉ๏Šฌ

Next, we use the multiplication rule for exponents, which states that ๐‘Žร—๐‘Ž=๐‘Ž,๏‰๏Š๏‰๏Šฐ๏Š where ๐‘Ž, ๐‘š, and ๐‘› are any real value.

We can then simplify the numerator and denominator, as some of the components have the same base: ๐‘šร—2๐‘š๐‘›ร—๐‘›=2ร—๐‘šร—๐‘š๐‘›ร—๐‘›=2๐‘š๐‘›=2๐‘š๐‘›.๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฌ๏Šฉ๏Šฌ๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฌ๏Šฉ๏Šฌ๏Šฑ๏Šฉ๏Šฑ๏Šฉ๏Šฐ๏Šฌ๏Šฉ๏Šฐ๏Šฌ๏Šฑ๏Šฉ๏Šฉ๏Šฏ

Lastly, we use the rule for negative exponents to simplify 2๏Šฑ๏Šฉ. This rule states that ๐‘Ž=1๐‘Ž,๏Šฑ๏Š๏Š where ๐‘Žโ‰ 0 and ๐‘› can be any real number.

So, we then have 2๐‘š๐‘›=2ร—๐‘š๐‘›=12ร—๐‘š๐‘›=18ร—๐‘š๐‘›=๐‘š8๐‘›.๏Šฑ๏Šฉ๏Šฉ๏Šฏ๏Šฑ๏Šฉ๏Šฉ๏Šฏ๏Šฉ๏Šฉ๏Šฏ๏Šฉ๏Šฏ๏Šฉ๏Šฏ

Therefore, ๏€ผ๐‘š๐‘›๏ˆ๏€พ2๐‘š๐‘›๏Š=๐‘š8๐‘›๏Šฑ๏Šง๏Šฑ๏Šฉ๏Šฑ๏Šจ๏Šฑ๏Šจ๏Šฑ๏Šฉ๏Šฉ๏Šฏ.

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

Letโ€™s consider the expression ๐‘ฅ๏Šจ.

We know that โˆš๐‘ฅ=๐‘ฅ,๐‘ฅโ‰ฅ0๏Šจfor and similarly ๏€บโˆš๐‘ฅ๏†=๐‘ฅ,๐‘ฅโ‰ฅ0.๏Šจfor

Letโ€™s suppose there is an index ๐‘š such that ๏€น๐‘ฅ๏…=๐‘ฅ.๏Šจ๏‰

By the power law for exponents, we know that ๏€น๐‘ฅ๏…=๐‘ฅ.๏Šจ๏‰๏Šจ๏‰

We also know that ๐‘ฅ=๐‘ฅ.๏Šง

Therefore, ๐‘ฅ=๐‘ฅ.๏Šจ๏‰๏Šง

So, by equating the indices, we get 2๐‘š=1,๐‘š=12.

So, ๏€น๐‘ฅ๏…=๐‘ฅ,๐‘ฅโ‰ฅ0.๏Šจ๏Ž ๏Žกfor

Since โˆš๐‘ฅ=๐‘ฅ๏Šจ, then we can deduce that ๏€น๐‘ฅ๏…=โˆš๐‘ฅ,๐‘ฅโ‰ฅ0.๏Šจ๏Šจ๏Ž ๏Žกfor

Therefore, we can see that any number ๐‘Žโ‰ฅ0 raised to the power of a half is equal to the square root of ๐‘Ž. In other words, ๐‘Ž=โˆš๐‘Ž,๐‘Žโ‰ฅ0.๏Ž ๏Žกfor

We can then follow the same steps for the general ๐‘›th root in the equation ๐‘Ž=โˆš๐‘Ž๏Š๏‘ƒ.

We know that for any positive integer ๐‘›: ๏€บโˆš๐‘ฅ๏†=๐‘ฅ,๐‘ฅโ‰ฅ0,๏‘ƒ๏Šfor and similarly ๏‘ƒโˆš๐‘ฅ=๐‘ฅ,๐‘ฅโ‰ฅ0.๏Šfor

Letโ€™s suppose there is an index ๐‘š such that (๐‘ฅ)=๐‘ฅ๐‘ฅโ‰ฅ0.๏Š๏‰for

By the power law for exponents we know that (๐‘ฅ)=๐‘ฅ.๏Š๏‰๏‰๏Š

We also know that ๐‘ฅ=๐‘ฅ.๏Šง

Therefore, ๐‘ฅ=๐‘ฅ.๏‰๏Š๏Šง

So, by equating the indices, we get ๐‘š๐‘›=1,๐‘š=1๐‘›.

So, (๐‘ฅ)=๐‘ฅ,๐‘ฅโ‰ฅ0.๏Š๏Ž ๏‘ƒfor

Since ๏‘ƒโˆš๐‘ฅ=๐‘ฅ๏Š, then we can deduce that (๐‘ฅ)=โˆš๐‘ฅ,๐‘ฅโ‰ฅ0.๏Š๏Š๏Ž ๏‘ƒ๏‘ƒfor

Therefore, we can see that any number ๐‘Žโ‰ฅ0 raised to the power of 1๐‘› is equal to the ๐‘›th root of ๐‘Ž. In other words, ๐‘Ž=โˆš๐‘Ž,๏Ž ๏‘ƒ๏‘ƒ for any positive integer ๐‘› and ๐‘Žโ‰ฅ0.

Having deduced the general ๐‘›th root, we can next use the law of exponents for powers to find a law for ๏€บโˆš๐‘Ž๏†๏‘ƒ๏‰.

We know the law of exponents for powers is (๐‘Ž)=๐‘Ž,๏‰๏Š๏‰๏Š where ๐‘Žโ‰ฅ0, and ๐‘š and ๐‘› can take any value in the real domain.

So, using ๐‘Ž=โˆš๐‘Ž๏Ž ๏‘ƒ๏‘ƒ, we can write ๏€บโˆš๐‘Ž๏†=๏€ฝ๐‘Ž๏‰=๐‘Ž=๐‘Ž,๏‘ƒ๏Ž ๏‘ƒ๏Ž ๏‘ƒ๏‘‚๏‘ƒ๏‰๏‰ร—๏‰ where ๐‘Žโ‰ฅ0, ๐‘› is a positive integer, and ๐‘š 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:

  • ๐‘Ž=โˆš๐‘Ž๏Ž ๏‘ƒ๏‘ƒ, for any value of ๐‘Žโ‰ฅ0 and any positive integer ๐‘›,
  • ๐‘Ž=๏€บโˆš๐‘Ž๏†=โˆš๐‘Ž๏‘‚๏‘ƒ๏‘ƒ๏‘ƒ๏‰๏‰, for any value of ๐‘Žโ‰ฅ0 and any positive integer ๐‘›.

Note, that in the laws above that, ๐‘Ž 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 ๐‘Ž 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 ๏€พ๐‘ฅ๐‘ฆ๏Š๏Šฎ๏Šฑ๏Šช๏Ž ๏Žก.

Answer

In order to simplify, we start with the power law for fractions, which states that ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏‰๏‰๏‰ where ๐‘โ‰ 0, and ๐‘Ž and ๐‘š can take any real value.

So, we have ๏€พ๐‘ฅ๐‘ฆ๏Š=๐‘ฅ๐‘ฆ.๏Šฎ๏Šฑ๏Šช๏Šฎร—๏Šฑ๏Šชร—๏Ž ๏Žก๏Ž ๏Žก๏Ž ๏Žก

So, we then get ๐‘ฅ๐‘ฆ=๐‘ฅ๐‘ฆ.๏Šฎร—๏Šฑ๏Šชร—๏Šช๏Šฑ๏Šจ๏Ž ๏Žก๏Ž ๏Žก

Next, we use the law of exponents for negative indices, which states that ๐‘Ž=1๐‘Ž,๐‘Žโ‰ 0.๏Šฑ๏Š๏Šwhere

This gives us ๐‘ฅ๐‘ฆ=๐‘ฅ=๐‘ฅรท1๐‘ฆ=๐‘ฅร—๐‘ฆ1=๐‘ฅ๐‘ฆ.๏Šช๏Šฑ๏Šจ๏Šช๏Šง๏˜๏Šช๏Šจ๏Šช๏Šจ๏Šช๏Šจ๏Žก

Therefore, ๏€พ๐‘ฅ๐‘ฆ๏Š=๐‘ฅ๐‘ฆ๏Šฎ๏Šฑ๏Šช๏Šช๏Šจ๏Ž ๏Žก.

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

Example 5: Simplifying Expressions with Negative Fractional Indices

Simplify the expression ๏๐‘ก๐‘ฃ๐‘ก๐‘ฃ๏๏Žข๏Žง๏Žช๏Žค๏Žฃ๏Žก๏Žข๏Ž ๏Žก๏Žช๏Žก๏Žข.

Answer

In order to simplify we start with the power law for fractions, which states that ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘,๏‰๏‰๏‰ where ๐‘โ‰ 0, and ๐‘Ž and ๐‘š can take any real value.

So, we have ๏๐‘ก๐‘ฃ๐‘ก๐‘ฃ๏=๏€ฝ๐‘ก๐‘ฃ๏‰๏€ฝ๐‘ก๐‘ฃ๏‰.๏Žข๏Žง๏Žช๏Žค๏Žฃ๏Žก๏Žข๏Ž ๏Žก๏Žช๏Žก๏Žข๏Žข๏Žง๏Žช๏Žค๏Žฃ๏Žช๏Žก๏Žข๏Žก๏Žข๏Ž ๏Žก๏Žช๏Žก๏Žข

Next, we apply the power law for products to the numerator and denominator, which states that (๐‘Ž๐‘)=๐‘Ž๐‘,๏‰๏‰๏‰ where ๐‘Ž, ๐‘, and ๐‘š can take any real value.

For the numerator, this gives us ๏€ฝ๐‘ก๐‘ฃ๏‰=๐‘กร—๐‘ฃ=๐‘กร—๐‘ฃ=๐‘กร—๐‘ฃ=๐‘กร—๐‘ฃ.๏Žข๏Žง๏Žช๏Žค๏Žฃ๏Žช๏Žก๏Žข๏Žข๏Žง๏Žช๏Žก๏Žข๏Žช๏Žค๏Žฃ๏Žช๏Žก๏Žข๏Žขร—๏Žช๏Žก๏Žงร—๏Žข๏Žช๏Žคร—๏Žช๏Žก๏Žฃร—๏Žข๏Žช๏Žฅ๏Žก๏Žฃ๏Ž ๏ŽŸ๏Ž ๏Žก๏Žช๏Ž ๏Žฃ๏Žค๏Žฅร—๏€ป๏‡ร—๏€ป๏‡

And for the denominator, we have ๏€ฝ๐‘ก๐‘ฃ๏‰=๐‘กร—๐‘ฃ=๐‘กร—๐‘ฃ=๐‘กร—๐‘ฃ=๐‘กร—๐‘ฃ.๏Žก๏Žข๏Ž ๏Žก๏Žช๏Žก๏Žข๏Žก๏Žข๏Žช๏Žก๏Žข๏Ž ๏Žก๏Žช๏Žก๏Žข๏Žกร—๏Žช๏Žก๏Žขร—๏Žข๏Ž ร—๏Žช๏Žก๏Žกร—๏Žข๏Žช๏Žฃ๏Žจ๏Žช๏Žก๏Žฅ๏Žช๏Žฃ๏Žจ๏Žช๏Ž ๏Žขร—๏€ป๏‡ร—๏€ป๏‡

Substituting back into the original expression, we get ๏€ฝ๐‘ก๐‘ฃ๏‰๏€ฝ๐‘ก๐‘ฃ๏‰=๐‘กร—๐‘ฃ๐‘กร—๐‘ฃ.๏Žข๏Žง๏Žช๏Žค๏Žฃ๏Žช๏Žก๏Žข๏Žก๏Žข๏Ž ๏Žก๏Žช๏Žก๏Žข๏Žช๏Ž ๏Žฃ๏Žค๏Žฅ๏Žช๏Žฃ๏Žจ๏Žช๏Ž ๏Žข

Next, we use the division rule for exponents, which states that ๐‘Žรท๐‘Ž=๐‘Ž,๐‘Žโ‰ 0.๏‰๏Š๏‰๏Šฑ๏Šwhere

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

So, ๐‘ก๐‘ฃ๐‘ก๐‘ฃ=๐‘ก๐‘กร—๐‘ฃ๐‘ฃ.๏Žช๏Ž ๏Žฃ๏Žค๏Žฅ๏Žช๏Žฃ๏Žจ๏Žช๏Ž ๏Žข๏Žช๏Ž ๏Žฃ๏Žช๏Žฃ๏Žจ๏Žค๏Žฅ๏Žช๏Ž ๏Žข

Simplifying the part of the expression with the ๐‘ก-terms gives us ๐‘ก๐‘ก=๐‘กรท๐‘ก=๐‘ก=๐‘ก=๐‘ก=๐‘ก.๏Žช๏Ž ๏Žฃ๏Žช๏Žฃ๏Žจ๏Žช๏Ž ๏Žฃ๏Žช๏Žฃ๏Žจ๏Žช๏Ž ๏Žฃ๏Žช๏Žฃ๏Žจ๏Žช๏Ž ๏Žฃ๏Žฃ๏Žจ๏Žช๏Žจ๏Žฉ๏Ž ๏Žฅ๏Žข๏Žฅ๏Žฆ๏Žข๏Žฅ๏Šฑ๏€ป๏‡๏Šฐ

Similarly, simplifying the part of the expression with the ๐‘ฃ-terms gives us ๐‘ฃ๐‘ฃ=๐‘ฃรท๐‘ฃ=๐‘ฃ=๐‘ฃ=๐‘ฃ=๐‘ฃ.๏Žค๏Žฅ๏Žช๏Ž ๏Žข๏Žค๏Žฅ๏Žช๏Ž ๏Žข๏Žค๏Žฅ๏Žช๏Ž ๏Žข๏Žค๏Žฅ๏Ž ๏Žข๏Žค๏Žฉ๏Žก๏Žฅ๏Žฆ๏Žฅ๏Šฑ๏€ป๏‡๏Šฐ

Finally, substituting both parts back into the expression, we get ๐‘ก๐‘กร—๐‘ฃ๐‘ฃ=๐‘กร—๐‘ฃ=๐‘ก๐‘ฃ.๏Žช๏Ž ๏Žฃ๏Žช๏Žฃ๏Žจ๏Žค๏Žฅ๏Žช๏Ž ๏Žข๏Žฆ๏Žข๏Žฅ๏Žฆ๏Žฅ๏Žฆ๏Žข๏Žฅ๏Žฆ๏Žฅ

Therefore, ๏๐‘ก๐‘ฃ๐‘ก๐‘ฃ๏=๐‘ก๐‘ฃ๏Žข๏Žง๏Žช๏Žค๏Žฃ๏Žก๏Žข๏Ž ๏Žก๏Žช๏Žก๏Žข๏Žฆ๏Žข๏Žฅ๏Žฆ๏Žฅ.

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
    • ๐‘Žร—๐‘Ž=๐‘Ž๏‰๏Š๏‰๏Šฐ๏Š,
    • ๐‘Žรท๐‘Ž=๐‘Ž,๐‘Žโ‰ 0๏‰๏Š๏‰๏Šฑ๏Šwhere,
    • (๐‘Ž)=๐‘Ž๏‰๏Š๏‰๏Š,
    • (๐‘Ž๐‘)=๐‘Ž๐‘๏‰๏‰๏‰,
    • ๏€ป๐‘Ž๐‘๏‡=๐‘Ž๐‘๏‰๏‰๏‰, ๐‘โ‰ 0.
  • The law of exponents for negative indices is 1๐‘Ž=๐‘Ž,๐‘Žโ‰ 0.๏Š๏Šฑ๏Šwhere
  • The laws of exponents for fractional indices are
    • ๐‘Ž=โˆš๐‘Ž๏Ž ๏‘ƒ๏‘ƒ, for any value of ๐‘Ž and any positive integer ๐‘›,
    • ๐‘Ž=๏€บโˆš๐‘Ž๏†=โˆš๐‘Ž๏‘‚๏‘ƒ๏‘ƒ๏‘ƒ๏‰๏‰, for ๐‘Žโ‰ฅ0 and any positive integer ๐‘›.

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