In this explainer, we will learn how to evaluate algebraic expressions with variable exponents or bases.
Recall that in algebraic expressions of the form , is a real number called the base and is an integer called the exponent (or index, or power). We refer to these as exponential expressions. When describing in words, we say “ to the power of ,” or simply “ to the power ,” which is the mathematical way to describe copies of the number multiplied together. As an example, “ to the power of 4” means
Sometimes the base or exponent of an exponential expression is assigned a specific value, which means we can evaluate the expression at that given value. For instance, if asked to evaluate when , we substitute for the base to get
It is also possible to combine terms together to form exponential expressions in more than one variable, such as or . Here, we will focus on expressions of this type, with variable bases or exponents, and learn how to evaluate them at given values of the variables.
Example 1: Evaluating a Numerical Exponent Expression Involving Addition
If and , find the value of .
The exponential expression has two variable bases, and , and we must find its value when and . Substituting these values into our expression, we have
Remembering the correct order of operations, we need to work out the two products individually and then add the results together. Since the square root of a number multiplied by itself gives the original (base) number, we see that and , so
We deduce that if and , then .
If two exponential expressions have the same base or exponent, then it is possible to combine them together in various ways. We can simplify and evaluate such expressions using the laws of exponents (often referred to as the laws of indices). Since we will need to apply these laws to later examples, we give a brief summary of them below.
Law: Laws of Exponents
For nonzero real numbers and and nonzero integers and , these laws of exponents hold:
Observe that by using these laws, we can obtain an alternative method of solution for example 1, as follows.
Substituting the values and into the exponential expression gives
This time, by the rational exponent law , we have , which means
Next, the power of a power law implies , so which agrees with our original answer as expected.
Our next example involves another exponential expression with variable bases. In this case, we shall use the above laws to simplify the expression before substituting in the given values of the bases. In particular, these laws offer a means of solving problems that feature negative exponents.
Example 2: Evaluating a Numerical Exponent Expression Involving Subtraction, Multiplication, and Division
If and , find the value of .
Recall that is an exponential expression. It has the variable bases and , and we need to find its value when and .
To start, we can rewrite the expression as follows:
Notice that the last expression contains two negative exponents. Applying the negative exponent law to the term , we find . Similarly, applying the rearranged version to the term , we get . Our original expression can therefore be rewritten as
Before substituting the given values for and into this expression, we can use the rational index law to write as and as , so
Then, we apply the power of a power law in four places, and simplify, to get
Thus, if and , then .
Notice how every step of the above example involving the laws of exponents was justified by reference to the relevant law. It is important to give a full explanation of one’s reasoning so that the method is clear.
Example 3: Finding the Numerical Value for an Algebraic Expression at Specific Values Using Laws of Exponents with Negative Exponents
Calculate the numerical value of when and .
In this question, we have the exponential expression , which has the single numerical base and the variable exponents and . We need to calculate its value when and .
As both terms in this product have the same base, they can be combined together using the product law to get
Then, we substitute the values and , which gives
By the power of a quotient law , this simplifies to
The next step is to apply the negative exponent law to the term , giving . Similarly, applying the rearranged version to the term gives . We then have which is a fraction in its simplest form.
Note that above we could have simplified the exponential expression by another method, as follows. Applying the negative exponent law , we have
Dividing by a power of a fraction is the same as multiplying by that power turned upside down, that is, multiplying by its reciprocal. Therefore,
Finally, by the power of a quotient law , we have as before.
We conclude that when and , the numerical value of is .
Our next example involves two exponential equations with numerical bases and variable exponents. This time, we will need to use the laws of exponents to rewrite one base in terms of the other.
Example 4: Evaluating Algebraic Expressions by Solving Exponential Equations Using Laws of Exponents
Given that , determine the value of .
In this question, we have the two exponential equations
The exponential expressions on the left-hand sides have the numerical bases 2 and 8, respectively, as well as the variable exponents and . Our strategy will be to solve these equations for and ; we can then substitute into the expression and evaluate it.
In the second equation, notice that the base , a power of 2. This means we can rewrite the base 8 in terms of the base 2, so the second equation becomes
By the power of a power law , we have , so our two exponential equations are now
We see that both equations have the same right-hand side, and that they are identical except in their exponents. This implies that the exponents themselves must be equal, so . Therefore, we have reduced the two original equations to a single one:
To solve this equation, we need to find the power of 2 that equals 512. We can use a calculator and try different numbers; for instance, , which is too small. Examining some higher powers of 2, we have
Thus, the solution to the exponential equation is . Remember, we also have that , so substituting the value of gives . Dividing both sides of this equation by 3, we get .
Lastly, we can determine the value of the algebraic expression at and :
We conclude that when , the value of is 6.
Finally, the techniques discussed here can be used to answer word problems from other fields. For instance, geometric formulas for area and volume often involve lengths that are squared or cubed (resulting in terms with exponents 2 or 3), so geometry provides a natural context for applying these strategies.
Example 5: Solving a Word Problem by Rearranging and Evaluating an Algebraic Expression
The volume of a right circular cone is given by , where . If the volume of a right circular cone equals 462 cm3 and the radius of its base is 7 cm, find the height of the cone.
We have been given the formula for the volume of a cone,
Looking at the right-hand side, as and are just numbers, this is an exponential expression with the variable bases (the radius) and (the height).
Since we need to find the height of the cone, our first step is to take this formula and rearrange it to make the subject. Multiplying both sides by 3 gives and dividing through by , we get which is the same as . Now, we are ready to substitute the values , , and from the question, so
Rewriting the denominator as , we can divide the top and bottom by 7 to simplify it to . Thus,
The radius of the cone is given in centimetres, so its height is also in centimetres; the height of the cone is 9 cm.
Let us finish by recapping some key concepts from this explainer.
- The term “exponential expressions” refers to algebraic expressions of the form , where is a real number called the base and is an integer called the exponent (or index, or power). We can evaluate exponential expressions with variable exponents (or bases) at a given value of the exponent (or base).
- When evaluating more complex exponential expressions, we can use the laws of exponents
to simplify them as much as possible.
For nonzero real numbers and and nonzero integers and , these laws hold:
- These techniques can be applied to solve geometric and real-world word problems that require evaluating exponential expressions with variable exponents or bases.