In this explainer, we will learn how to evaluate real numbers raised to positive and negative integer and zero powers and solve simple exponential equations.
Recall that to evaluate a power, we multiply the base the number of times indicated by the exponent. Let’s recap the definition below.
Definition: Evaluating a Power
For a power with base , , and exponent , , then
Using this definition, we can evaluate not only powers where the base is an integer or a rational number, but also where the base is a real number.
One such type of real number we might evaluate is a square root. Recall that a square root is a number written in the form , . We can use the inverse operation, squaring, to derive two key results.
The first is that if you square , then you get , since squaring and square rooting are inverse operations (i.e., , ).
The second is that if you square root , you either get or , depending on whether is negative or positive. This is because when we square a negative number, say , we get a positive number, 9. However, if we square the additive inverse of , which is 3, we also get 9. This means when we square root a number squared, the result is positive, but the original number might have been negative. To avoid confusion, we then say that . However, if we know , then we can say .
Expanding on these definitions and using the definition of a power, we can say that and . Therefore, we get the following definition.
Definition: Simplifying Expressions with Squares and Square Roots
For an expression with a base , where ,
In our first example, we will apply what we know about evaluating powers and calculating squares of radicals to determine the value of an unknown.
Example 1: Raising a Real Number to a Positive Integer Power
Determine the value of given that .
To find the value of , we need to substitute into . Doing so gives us
We know from the definition of a power that is the same as saying . Therefore, we get
To calculate this, we need to use , giving us
Therefore, the value of is 640.
As well as raising real numbers to the power of a positive integer, we can also raise them to the power of zero or to the power of a negative integer.
If we consider the pattern of powers of 2, we can see that as we increase the power by 1, we multiply by 2, as follows:
If we move backward in the sequence, we can therefore deduce that must be divided by 2. This then gives us . So, .
Similarly, if we move backward again, we can deduce the must be divided by 2. This then gives us . So, .
Again, if we continue moving backward, we can deduce that and and so on. If we write out the sequence again, we get the following:
Notice, that if we rewrite the denominators in the first three terms as powers of 2, we get the following:
We can then say that , , and . Continuing the sequence in the negative direction, we can generalize that , .
These results do not only hold for a base of 2 but also for any real number except zero. We can generalize these in the laws below.
Laws: Rules for Zero and Negative Exponents
- The law for zero exponents is where .
- The law for negative exponents is where and .
In the following example, we will use the law for zero exponents to determine the value of an expression involving powers of radicals.
Example 2: Raising a Real Number to a Zero Power
If , find .
To determine the value of , we can substitute , giving us
We know that , where ; therefore,
Therefore, the value of is 1.
In the next example, we will use the law for negative exponents to determine the value of an expression involving powers of radicals.
Example 3: Raising a Real Number to a Negative Integer Power
If , find .
To find the value of when , we first need to substitute. This gives us
Next, to simplify , we use the law for negative exponents, which states that where and . This then gives us
We can use the rule , , to evaluate the denominator by expanding the power, giving us
Substituting this into the denominator, we get
Therefore, the value of when is .
In the next example, we will discuss how to evaluate an expression with positive and negative exponents.
Example 4: Evaluating an Expression by Substituting Surds
If , , and , then find the value of in its simplest form.
To find the value of , we start by substituting , , and . Doing so gives us
We can simplify this further by expanding the brackets of the second term, giving us
Canceling common factors of in gives us
Substituting this back into the expression, we get
Next, we can use to evaluate the squared terms, giving us for the first, second, and fourth terms, respectively,
Substituting back into the expression, we get
Since is , then we can use to simplify this, giving us
Substituting back into the expression, we get
To simplify further, we need to find a common factor for the denominators of the fractions. For the first four terms, a common factor is 36, giving us
Finally, a common factor of 36 and 81 is 324, giving us
Therefore, the answer is .
So far, we have considered how to simplify expressions with integer exponents. Next, we will discuss how to find an unknown exponent.
When comparing two powers with the same base, we can deduce that if the powers are equal, then their exponents must be equal as well. For example, if , then since the bases are the same, the exponents must be the same too, meaning .
Similarly, if we compare two powers with the same exponent, then if they are equal and odd, the bases must be equal. For example, if , then since the exponents are the same, then the bases must be the same too, meaning . However, if we have an even exponent, then it is possible that the bases are not equal, since one could be the negative of the other. For example, , so if the exponents are equal, then the base could be or 2 in order for them to be equal. This means that if the even exponents are equal, then we can say the moduli of the bases are equal. For example, if , then , so or . Let’s summarize these points in the rule below.
Rule: Equating Powers with the Same Base or the Same Exponent
- If , where , then .
- If , then when and when .
We can use these rules to solve problems where there are unknowns in the exponent. Let’s discuss how to do this in the following example.
Example 5: Solving a Simple Exponential Equation
Find the value of in the equation .
To find the value of in the equation , we can use the rule that states that if , where , then .
This rule will only work if both sides of the equation have the same base. As such, since the base of the power on the left-hand side of the equation is 3, then we want to write the right-hand with a base of 3 also.
We know that 81 is a power of 3, since . Therefore, . We can then say that the right-hand side is
Next, we can use the law of negative exponents, which states where and , meaning that for , we get
Putting this into the right-hand side of the original equation, we then get
Since the bases are now the same, we can equate the exponents, giving us
Solving for , we get
Therefore, the value of is for the equation .
In this explainer, we have learned how to evaluate real numbers raised to positive, zero, and negative exponents and how to solve simple exponential equations. Let’s recap the key points.
- We can use the property to simplify radicals raised to positive integers.
- We can use the law for zero exponents to simplify real numbers raised to the power of zero. The law states that where .
- We can use the law for negative exponents to simplify real numbers raised to negative powers. The law states that where and .
- We can use the rules for when bases or exponents are equal for equal powers to find unknowns in simple exponential equations. The rules state that
- if , where , then ,
- if , then when and when .