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

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: 𝑎×𝑎=𝑎, where 𝑚 and 𝑛 can take any value in the real domain.
  • Division of exponents with the same base: 𝑎÷𝑎=𝑎, where 𝑎0 and where 𝑚 and 𝑛 can take any value in the real domain.

Since 𝑚 and 𝑛 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, 𝑎÷𝑎=𝑎, where 𝑎0, we can see that when 𝑚<𝑛, this will result in a negative index.

Further, if we let 𝑚=0, we can see 𝑎÷𝑎=𝑎,𝑎0.where

Recall that 𝑎=1 when 𝑎0, so we get the following formula: 1÷𝑎=𝑎,𝑎0.where

Or, written as a fraction, we get the following: 1𝑎=𝑎,𝑎0.where

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: 𝑎=1𝑎, where 𝑎0 and 𝑛 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 𝑥, we would rewrite this as 1𝑥 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 109𝑥𝑦?

  1. 910𝑥𝑦
  2. 109𝑥𝑦
  3. 109𝑥𝑦
  4. 10𝑥𝑦9

Answer

In order to rewrite the expression 109𝑥𝑦, we must use the law of exponents for negative exponents, which states that 𝑎=1𝑎,𝑎0.where

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

For 𝑥, we can substitute 𝑛=2 and 𝑎=𝑥, which gives us 𝑥=1𝑥.

For 𝑦, we can substitute 𝑛=7 and 𝑎=𝑦, which gives us 𝑦=1𝑦.

Now we have rewritten both variables with positive exponents, we can then put them back into the original expression, giving us 109𝑥𝑦=109×1𝑥×1𝑦=10×1×19×𝑥×𝑦=109𝑥𝑦.

Hence, the answer is option C, 109𝑥𝑦.

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 𝑥𝑥 is 1𝑥.

Answer

In order to simplify 𝑥𝑥, we start by using the division rule for exponents: 𝑎÷𝑎=𝑎,𝑎0.where

In this case, even though both exponents are negative, with 𝑚=4 and 𝑛=2, we can still apply this rule. It also helps to rewrite the fraction as a division here. By doing so we get 𝑥𝑥=𝑥÷𝑥=𝑥=𝑥.()

Next, since the index is still negative, then we use the law of exponents for negative exponents, which states that 𝑎=1𝑎,𝑎0.where

By substituting 𝑛=2 and 𝑎=𝑥 into the law, we get 𝑥=1𝑥.

Therefore, 𝑥𝑥=1𝑥, so the answer is true.

Looking back at the previous example, there are other ways of simplifying 𝑥𝑥 to get 1𝑥.

Another approach is to use the law of exponents for negative indices first, which states that 𝑎=1𝑎,𝑎0.where

We then substitute 𝑛=4 and 𝑎=𝑥 for the term in the numerator and 𝑛=2 and 𝑎=𝑥 for the term in the denominator. This gives us 𝑥𝑥=.

We can then use our understanding of fractions to write this as =1𝑥÷1𝑥=1𝑥×𝑥=𝑥𝑥.

We then use the rule for division of exponents, which states that 𝑎÷𝑎=𝑎,𝑎0.where

With 𝑎=𝑥, 𝑚=2, and 𝑛=4, we get 𝑥𝑥=𝑥÷𝑥=𝑥=𝑥=1𝑥.

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

  • (𝑎)=𝑎,
  • (𝑎𝑏)=𝑎𝑏,
  • 𝑎𝑏=𝑎𝑏, 𝑏0,

where 𝑚 and 𝑛 can take any value in the real domain.

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

Simplify 𝑚𝑛2𝑚𝑛.

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 𝑎𝑏=𝑎𝑏, where 𝑏0 and 𝑚 can take any real value.

When applying this law to the first part of the expression, 𝑚𝑛, we get 𝑚𝑛=𝑚(𝑛).

We can simplify the denominator further using the law of exponents for powers, that states: (𝑎)=𝑎, where 𝑚 and 𝑛 can take any real value.

So, we get 𝑚(𝑛)=𝑚𝑛=𝑚𝑛.×

Similarly, we can apply the law for powers of fractions to the second part of the expression, 2𝑚𝑛, which gives us. 2𝑚𝑛=2𝑚(𝑛).

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 (𝑎𝑏)=𝑎𝑏, where 𝑚 can take any real value.

So, we get 2𝑚(𝑛)=2×𝑚𝑛=2×𝑚𝑛=2×𝑚𝑛.×××

When combining the two parts of the expression, 𝑚𝑛=𝑚𝑛2𝑚𝑛=2×𝑚𝑛,and we get 𝑚𝑛2𝑚𝑛=𝑚𝑛×2𝑚𝑛=𝑚×2𝑚𝑛×𝑛.

Next, we use the multiplication rule for exponents, which states that 𝑎×𝑎=𝑎, where 𝑎, 𝑚, and 𝑛 are any real value.

We can then simplify the numerator and denominator, as some of the components have the same base: 𝑚×2𝑚𝑛×𝑛=2×𝑚×𝑚𝑛×𝑛=2𝑚𝑛=2𝑚𝑛.

Lastly, we use the rule for negative exponents to simplify 2. This rule states that 𝑎=1𝑎, where 𝑎0 and 𝑛 can be any real number.

So, we then have 2𝑚𝑛=2×𝑚𝑛=12×𝑚𝑛=18×𝑚𝑛=𝑚8𝑛.

Therefore, 𝑚𝑛2𝑚𝑛=𝑚8𝑛.

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

Let’s consider the expression 𝑥.

We know that 𝑥=𝑥,𝑥0for and similarly 𝑥=𝑥,𝑥0.for

Let’s suppose there is an index 𝑚 such that 𝑥=𝑥.

By the power law for exponents, we know that 𝑥=𝑥.

We also know that 𝑥=𝑥.

Therefore, 𝑥=𝑥.

So, by equating the indices, we get 2𝑚=1,𝑚=12.

So, 𝑥=𝑥,𝑥0.for

Since 𝑥=𝑥, then we can deduce that 𝑥=𝑥,𝑥0.for

Therefore, we can see that any number 𝑎0 raised to the power of a half is equal to the square root of 𝑎. In other words, 𝑎=𝑎,𝑎0.for

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

We know that for any positive integer 𝑛: 𝑥=𝑥,𝑥0,for and similarly 𝑥=𝑥,𝑥0.for

Let’s suppose there is an index 𝑚 such that (𝑥)=𝑥𝑥0.for

By the power law for exponents we know that (𝑥)=𝑥.

We also know that 𝑥=𝑥.

Therefore, 𝑥=𝑥.

So, by equating the indices, we get 𝑚𝑛=1,𝑚=1𝑛.

So, (𝑥)=𝑥,𝑥0.for

Since 𝑥=𝑥, then we can deduce that (𝑥)=𝑥,𝑥0.for

Therefore, we can see that any number 𝑎0 raised to the power of 1𝑛 is equal to the 𝑛th root of 𝑎. In other words, 𝑎=𝑎, for any positive integer 𝑛 and 𝑎0.

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

We know the law of exponents for powers is (𝑎)=𝑎, where 𝑎0, and 𝑚 and 𝑛 can take any value in the real domain.

So, using 𝑎=𝑎, we can write 𝑎=𝑎=𝑎=𝑎,× where 𝑎0, 𝑛 is a positive integer, and 𝑚 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:

  • 𝑎=𝑎, for any value of 𝑎0 and any positive integer 𝑛,
  • 𝑎=𝑎=𝑎, for any value of 𝑎0 and any positive integer 𝑛.

Note, that in the laws above that, 𝑎 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 𝑎 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 𝑥𝑦.

Answer

In order to simplify, we start with the power law for fractions, which states that 𝑎𝑏=𝑎𝑏, where 𝑏0, and 𝑎 and 𝑚 can take any real value.

So, we have 𝑥𝑦=𝑥𝑦.××

So, we then get 𝑥𝑦=𝑥𝑦.××

Next, we use the law of exponents for negative indices, which states that 𝑎=1𝑎,𝑎0.where

This gives us 𝑥𝑦=𝑥=𝑥÷1𝑦=𝑥×𝑦1=𝑥𝑦.

Therefore, 𝑥𝑦=𝑥𝑦.

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

Example 5: Simplifying Expressions with Negative Fractional Indices

Simplify the expression 𝑡𝑣𝑡𝑣.

Answer

In order to simplify we start with the power law for fractions, which states that 𝑎𝑏=𝑎𝑏, where 𝑏0, and 𝑎 and 𝑚 can take any real value.

So, we have 𝑡𝑣𝑡𝑣=𝑡𝑣𝑡𝑣.

Next, we apply the power law for products to the numerator and denominator, which states that (𝑎𝑏)=𝑎𝑏, where 𝑎, 𝑏, and 𝑚 can take any real value.

For the numerator, this gives us 𝑡𝑣=𝑡×𝑣=𝑡×𝑣=𝑡×𝑣=𝑡×𝑣.×××××()×()

And for the denominator, we have 𝑡𝑣=𝑡×𝑣=𝑡×𝑣=𝑡×𝑣=𝑡×𝑣.×××××()×()

Substituting back into the original expression, we get 𝑡𝑣𝑡𝑣=𝑡×𝑣𝑡×𝑣.

Next, we use the division rule for exponents, which states that 𝑎÷𝑎=𝑎,𝑎0.where

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

So, 𝑡𝑣𝑡𝑣=𝑡𝑡×𝑣𝑣.

Simplifying the part of the expression with the 𝑡-terms gives us 𝑡𝑡=𝑡÷𝑡=𝑡=𝑡=𝑡=𝑡.()

Similarly, simplifying the part of the expression with the 𝑣-terms gives us 𝑣𝑣=𝑣÷𝑣=𝑣=𝑣=𝑣=𝑣.()

Finally, substituting both parts back into the expression, we get 𝑡𝑡×𝑣𝑣=𝑡×𝑣=𝑡𝑣.

Therefore, 𝑡𝑣𝑡𝑣=𝑡𝑣.

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
    • 𝑎×𝑎=𝑎,
    • 𝑎÷𝑎=𝑎,𝑎0where,
    • (𝑎)=𝑎,
    • (𝑎𝑏)=𝑎𝑏,
    • 𝑎𝑏=𝑎𝑏, 𝑏0.
  • The law of exponents for negative indices is 1𝑎=𝑎,𝑎0.where
  • The laws of exponents for fractional indices are
    • 𝑎=𝑎, for any value of 𝑎 and any positive integer 𝑛,
    • 𝑎=𝑎=𝑎, for 𝑎0 and any positive integer 𝑛.

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