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

In this explainer, we will learn how to use the rules of negative and fractional indices to simplify numerical expressions.

Recall that an exponential is an expression of the form π‘ŽοŒ». We refer to the number π‘Ž in such an expression as the base of the exponential and to 𝑏 as the power, exponent, or index. When 𝑏 is a positive whole number, the expression π‘ŽοŒ» is straightforward to interpret: it means 𝑏 lots of π‘Ž, all multiplied together. For example, 3=3Γ—3Γ—3Γ—3Γ—3.

Throughout this explainer, we will interpret expressions of the form π‘ŽοŒ» when 𝑏 is not a positive whole number, that is, when 𝑏 is negative, a positive fraction, or even a negative fraction.

First of all, let us recall the laws of exponents.

Laws: The Laws of Exponents

For all numbers π‘Ž, 𝑏, and 𝑐, we have three laws of exponents:

  1. The product rule of exponents: π‘ŽΓ—π‘Ž=π‘ŽοŒ»οŒΌοŒ»οŠ°οŒΌ
  2. The quotient rule of exponents: π‘ŽΓ·π‘Ž=π‘ŽοŒ»οŒΌοŒ»οŠ±οŒΌ
  3. The power rule of exponents: ο€Ήπ‘Žο…=π‘ŽοŒ»οŒΌοŒ»οŒΌ

Now, consider the following expression: 5Γ·5.οŠͺ

We can evaluate this directly as shown: 5Γ·5=(5Γ—5Γ—5)Γ·(5Γ—5Γ—5Γ—5)=5Γ—5Γ—55Γ—5Γ—5Γ—5.οŠͺ

By canceling common factors in the numerator and denominator, we have that 5Γ·5=15.οŠͺ

However, we can also evaluate 5Γ·5οŠͺ using the laws of indices: 5Γ·5=5=5,οŠͺοŠͺ which leads us to the equation 5=15.

Of course, nothing depends on the base, which is 5 here, and a little experimentation with other bases should convince you that the same argument will yield the equation π‘Ž=1π‘Ž, for any base π‘Ž.

Now, try the same argument again with indices that differ by more than one, for example 3Γ·3οŠͺ. By considering a few such examples, we should be able to convince ourselves that a very compelling interpretation of the expression π‘ŽοŠ±οŒ» is π‘Ž=1π‘Ž.

Indeed, we will take this as a definition.

Definition: Negative Exponents

For any numbers π‘Žβ‰ 0 and 𝑏, we define π‘Ž=1π‘Ž.

It is worth pausing to note a special case of this definition, namely when π‘Ž is a fraction, say π‘Ž=𝑐𝑑. In this case, we have 𝑐𝑑=𝑐𝑑=𝑐𝑑,οŠ±οŒ»οŠ±οŠ§Γ—οŒ»οŠ±οŠ§οŒ» which, by the definition of negative exponents, is equal to 1.

Since dividing by a fraction is the same thing as multiplying by its reciprocal, we have 𝑐𝑑=1=𝑑𝑐.

That is to say taking a negative power of a fraction has the effect of β€œflipping” the fraction.

Example 1: Evaluating the Negative Exponent of a Number

Express 4 in the form π‘Žπ‘, where π‘Ž and 𝑏 are integers.

Answer

Recall that for any numbers π‘Žβ‰ 0 and 𝑏, we define π‘Ž=1π‘Ž.

Then, by definition, we have 4=14=164.

Thus, we have π‘Ž=1, 𝑏=64, and, for our answer, 4=164.

Now, let us turn our attention to fractional exponents. What are we to make, for example, of the expression 4?

Well, let us try squaring it: ο€½4.

We can now apply the laws of exponents, specifically the power rule ο€Ήπ‘Žο…=π‘ŽοŒ»οŒΌοŒ»οŒΌ. Thus, ο€½4=4=4=4.οŽ οŽ‘οŽ οŽ‘οŠ¨Γ—οŠ¨οŠ§

Now, ask yourself the following: What is the equation ο€½4=4 saying? It is saying that this number, 4, whatever it is, has the property that when you square it, you get 4. This statement is nothing other than the definition of a square root of 4. Hence, we arrive at the interpretation 4=√4.

Be aware that 4 (like all positive real numbers) has not one but two square roots, namely +2 and βˆ’2. Nothing we have said so far favors one of these over the other as the interpretation of 4. In order for 4 to be well defined, we follow the convention that π‘ŽοŽ οŽ‘ denotes the principal (in this case, positive) square root of π‘Ž.

Let us try another expression: 10, but this time, we are going to cube it: ο€½10=10=10=10.οŽ οŽ’οŽ οŽ’οŠ©Γ—οŠ©οŠ§

The equation ο€½10=10 is now telling us that the number 10 (whatever it is) has the property that when you cube it, you get 10. In other words, 10 is a cube root of 10: 10=√10.

A little further experimentation should convince you that a sensible interpretation for the expression π‘ŽοŽ ο΄ is ο΄βˆšπ‘Ž. Indeed, we will take this as a definition.

Definition: Fractional Exponents with Numerator 1

For any numbers π‘Ž and 𝑏, we define π‘Ž=βˆšπ‘Ž, where ο΄βˆšπ‘Ž denotes the principal 𝑏th root of π‘Ž, with the understanding that when 𝑏 is even and π‘Ž is negative, this expression is not defined over the real numbers.

We now have an interpretation for expressions of the form π‘ŽοŽ ο΄, that is, fractional exponents in which the numerator of the exponent is 1. Once again, we can use the laws of exponents to guide us here. Consider the expression 27. By the power rule of exponents, ο€Ήπ‘Žο…=π‘ŽοŒ»οŒΌοŒ»οŒΌ, we have 27=27=ο€Ή27,οŽ‘οŽ’οŽ οŽ’οŽ οŽ’οŠ¨Γ—οŠ¨ to which we can apply the previous definition: 27=ο€Ή27=√27.

Notice that we can multiply the factors in the exponent (in this example, 2 and 13) in either order. That is, 27=27=ο€Ή27=√2727=27=ο€½27=ο€»βˆš27.οŽ‘οŽ’οŽ οŽ’οŽ οŽ’οŽ’οŽ‘οŽ’οŽ οŽ’οŽ οŽ’οŽ’οŠ¨Γ—οŠ¨οŠ¨Γ—οŠ¨οŠ¨οŠ¨or

Definition: General Fractional Exponents

For any numbers π‘Ž, 𝑏, and 𝑐, we define π‘Ž=βˆšπ‘Ž=(βˆšπ‘Ž), where ο΅βˆšπ‘Ž denotes the principal 𝑐th root of π‘Ž, with the understanding that when 𝑐 is even and π‘Ž is negative, this expression may not be defined over the real numbers.

The important point here is that in evaluating an expression of the form π‘Žο΄ο΅, the order of the operations of the root and exponent makes no difference. That is, ο΅ο΅βˆšπ‘Ž=ο€Ίβˆšπ‘Žο†οŒ»οŒ». In practical calculations, you will often find that it is easier to compute the root before the exponent.

Example 2: Evaluating the Fractional Exponent of a Number

Evaluate 8.

Answer

Recall that for any numbers π‘Ž, 𝑏, and 𝑐, we define π‘Ž=βˆšπ‘Ž=ο€Ίβˆšπ‘Žο†, where ο΅βˆšπ‘Ž denotes the principal 𝑐th root of π‘Ž.

Then, by definition, we have 8=√8=ο€»βˆš8.

Now, we have two options for completing the calculation: we can either compute the cube root of 8 or the square of √8, since the two are equivalent. It is usually easier to compute the root before the exponent, simply because calculating powers of smaller numbers is generally easier than calculating roots of bigger ones. Thus, by squaring first, we have 8=√8=√64=4, or, by taking the cube root first, we have 8=ο€»βˆš8=2=4.

Either option gives us the answer 8=4.

Be aware that a fractional exponent may sometimes appear in disguised form, that is to say, as a decimal. One should not be afraid of such expressions as 9,125,16.οŠ¦οŽ–οŠ«οŠ¦οŽ–Μ‡οŠ©οŠ§οŽ–οŠ¨οŠ«and

In this situation, simply convert the decimal into a fraction and continue as normal. Thus, 9=9=√9=3,125=125=√125=5,16=16=ο€»βˆš16=2=32.οŠ¦οŽ–οŠ«οŠ¦οŽ–Μ‡οŠ©οŠ§οŽ–οŠ¨οŠ«οŠ«οŠ«οŽ οŽ‘οŽ οŽ’οŽ’οŽ€οŽ£οŽ£

Example 3: Evaluating the Fractional Exponent of a Fraction

Given that ο€Ό827=π‘₯π‘¦οŽ£οŽ’, where π‘₯ and 𝑦 are integers, find π‘₯ and 𝑦.

Answer

Recall that for any numbers π‘Ž, 𝑏, and 𝑐, we define π‘Ž=βˆšπ‘Ž=ο€Ίβˆšπ‘Žο†, where ο΅βˆšπ‘Ž denotes the principal 𝑐th root of π‘Ž.

Then, by definition, we have ο€Ό827=ο€Ώο„ž827.οŠͺ

To proceed, we use the distributive property of roots over division, that is, the law that ο‘ƒο‘ƒο‘ƒο„žπ‘Žπ‘=βˆšπ‘Žβˆšπ‘, for all π‘Ž, 𝑏, and 𝑐. So, ο€Ό827=ο€Ό827=ο€Ώο„ž827=ο€Ώβˆš8√27=ο€Ό23.οŽ£οŽ’οŽ οŽ’οŽ’οŽ’οŽ’Γ—οŠͺοŠͺοŠͺοŠͺ

Completing the evaluation by raising everything to the power of 4, we have ο€Ό827=ο€Ό23=1681.οŠͺ

Thus, ο€Ό827=1681=π‘₯π‘¦οŽ£οŽ’, and so our answer is π‘₯=16 and 𝑦=81.

We are now going to combine both negative and fractional exponents in a single calculation. Suppose we are asked to evaluate the expression (125).

The first thing to do is to look at the exponent, which consists of up to three parts: a numerator, a denominator, and a sign (+/βˆ’).

So, in evaluating this example, (125), we will first take the cube root of 125, we will then square the result, and we will finally take the reciprocal of what we end up with, or in symbols, (125)=ο€»βˆš125=5=ο€Ή5=25=125.

Example 4: Evaluating the Negative Fractional Exponent of a Number

Evaluate 27.

Answer

Recall that for any numbers π‘Ž, 𝑏, and 𝑐, we define π‘Ž=βˆšπ‘Ž=ο€Ίβˆšπ‘Žο†, where ο΅βˆšπ‘Ž denotes the principal 𝑐th root of π‘Ž, and recall that for any numbers π‘Žβ‰ 0 and 𝑏, we define π‘Ž=1π‘Ž.

Looking at the exponent in the expression 27, we notice two things: a 3 in the denominator, telling us that we need to take a cube root of the base, and a negative sign, telling us to take the reciprocal as a final step. Thus, using the power rule of exponents, ο€Ήπ‘Žο…=π‘ŽοŒ»οŒΌοŒ»οŒΌ, 27=27=ο€½27.οŠ±Γ—οŠ±οŠ§οŠ±οŠ§οŽ οŽ’οŽ οŽ’οŽ οŽ’

The fraction 13 in the exponent is telling us to take a cube root, =ο€»βˆš27=3, and finally take the reciprocal, =13, giving us a final answer of 13.

Our last example will combine everything we have covered in this explainer so far.

Example 5: Evaluating the Negative Fractional Exponent of a Fraction Using Laws of Indices

Evaluate ο€Ύο€Ό6416.

Answer

Recall that for any numbers π‘Ž, 𝑏, and 𝑐, we define π‘Ž=βˆšπ‘Ž=ο€Ίβˆšπ‘Žο†, where ο΅βˆšπ‘Ž denotes the principal 𝑐th root of π‘Ž, and recall that for any numbers π‘Žβ‰ 0 and 𝑏, we define π‘Ž=1π‘Ž.

There is more than one way to approach this question. The temptation is perhaps to cube the fraction 6416 first and then tackle the exponent of βˆ’12. However, if you do not have a calculator handy, this could be a bit of a challenge. Remember that we can often use the laws of exponents to rearrange the order of calculations, on the basis that in calculations like this, it is usually easier to take roots before powers, saving taking the reciprocal until last. In particular, we will use the power rule of exponents, ο€Ήπ‘Žο…=π‘ŽοŒ»οŒΌοŒ»οŒΌ. So, ο€Ύο€Ό6416=ο€Ό6416.(οŠ©Γ—οŠ±)

Since multiplication is commutative, we can change the order of operations in the exponent: =ο€Ό6416.(Γ—οŠ±οŠ©)

Applying the power rule, we have =6416.

Now, we can take the square root of the innermost parentheses, 6416=ο€Ώο„ž6416=ο€Ό84=2, and apply the power rule of exponents again, 2=2=ο€Ή2=8.οŠ±οŠ©οŠ©Γ—οŠ±οŠ§οŠ©οŠ±οŠ§οŠ±οŠ§

Finally, we take the reciprocal: 8=18.

Therefore, our final answer is ο€Ύο€Ό6416=18.

Let us finish by recapping a few important concepts from this explainer.

Key Points

  • The general formula for negative exponents is given by π‘Ž=1π‘Ž, for any numbers π‘Ž and 𝑏.
  • The general formula for fractional exponents is given by π‘Ž=βˆšπ‘Ž=ο€Ίβˆšπ‘Žο†, for any numbers π‘Ž, 𝑏, and 𝑐.
  • When an expression contains some combination of different, possibly negative, possibly fractional exponents, we can often use the index law ο€Ήπ‘Žο…=π‘ŽοŒ»οŒΌοŒ»οŒΌ to rearrange the order of operations. It is usually (though not always) easier to compute roots (that is, exponents of the form π‘ŽοŽ ο΄) first and save taking reciprocals (that is, exponents of the form π‘ŽοŠ±οŠ§) until the end.

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