In this explainer, we will learn how to apply the laws of exponents to multiply and divide powers and work out a power raised to a power.
We begin by recalling that repeated multiplication is represented by exponentiation. In general, we say that raising a number to a positive integer exponent is the same as multiplying it by itself such that it appears times in the product. If we have a base of , then we have the following:
This definition and result allow us to prove some useful results in evaluating exponential expressions. We call these the laws of exponents.
First, we can consider multiplying two exponential expressions with the same base. For example, . We have
This suggests that when we multiply exponential expressions with the same base, we add the exponents. We can show that this result holds in general by following the same process.
Letโs say that and and are positive integers; then,
Remember we can evaluate a product of rational numbers in any order, so we can rewrite this as a product of with itself where it appears times. Hence,
We can follow a similar process if we are dividing two exponential expressions with the same base. However, we also need the exponent of the dividend to be greater than the exponent of the divisor.
For example, if we want to calculate , we have
Instead of dividing, we multiply by the reciprocal to get
We then write out the exponents in full and cancel out the shared factors:
Since we cancel out four of the factors of 2 and 3 from the powers of 5, we can think of this as . This then suggests that when we divide exponential expressions with the same base, we subtract the exponents. We can show that this result holds for any rational number by following the same process.
If and and are positive integers with , we have
We can rearrange by writing the products out in full to get
Since , we can cancel out of the shared factors of in the first fraction and of the shared factors of in the second fraction. This will leave factors of in the numerator and factors of in the denominator. Hence,
Finally, we can rewrite this as an exponent of . We have
There is one last rule we can deduce by using this definition. We can consider what would happen if we raised a base to multiple exponents.
For example, letโs evaluate . We start with the expression inside the parentheses, so
We then need to cube this expression. So, we get
We see that this is 4 multiplied by itself, where it appears 6 times in the product. However, we can see where this value of 6 comes from. Since we cube , we multiply the number of factors by 3, which gives us factors of 4. Hence, we have
We can show that this result holds for any rational number by following the same process.
If and and are positive integers, then
We can summarize all the results we have just proven as follows.
Rules: Laws of Exponents for Rational Bases
If and and are positive integers, then
- ,
- ,
- ,
- and if , then
.
Letโs now see some examples of using the laws of exponents to evaluate and simplify exponential expressions.
Example 1: Multiplying Negative Mixed Number Expressions with Positive Integer Exponents
Which of the following is equal to ?
Answer
We could evaluate this expression by evaluating each factor separately; however, we can also simplify the expression first by using the laws of exponents. We recall that if we multiply two powers with the same base, then we can just add the exponents, for any rational number and positive integers and .
So,
We can evaluate this expression by first rewriting as a fraction. We note that . Thus,
Finally, we recall that if is a rational number and is a positive integer, then .
Hence,
We could leave our answer like this; however, it is not one of the options. Instead, we rewrite this as a mixed number. We have which is option B.
In our next example, we will use a different one of the laws of exponents to evaluate an expression.
Example 2: Dividing Negative Mixed Number Expressions by Positive Integer Exponents
Which of the following has the same value as ?
Answer
We could evaluate the dividend and divisor separately and then evaluate the division. However, it is easier to simplify the expression first. We recall that the laws of exponents tell us that when dividing powers with the same base, we subtract the exponents, for any rational number and positive integers and with .
Thus,
We then recall that raising a number to an exponent of 1 leaves it unchanged, so
Finally, we convert our answer into a fraction. We have which is option D.
In our next example, we will determine which of a set of given options is equivalent to a given exponential expression.
Example 3: Evaluating Positive Rational Expressions with Positive Integer Exponents Raised to Positive Integer Exponents
Which of the following expressions has the same value as ?
Answer
We could evaluate this expression and all of the options by just evaluating the exponents. However, it is easier to simplify the given expression by using the laws of exponents.
We recall that if we raise a rational base to a positive integer exponent and then raise this to another positive integer exponent, then we can simplify by multiplying the exponents, for any rational number and positive integers and .
Therefore,
We see that this is option A.
In our next example, we will use the laws of exponents to simplify the evaluation of a given exponential expression.
Example 4: Evaluating a Numerical Expression Using the Laws of Exponents
Calculate , giving your answer in its simplest form.
Answer
We could evaluate this expression by evaluating each power in turn and then multiplying the fractions. However, this is a difficult process and it would be easy to make mistakes. Instead, we can simplify by using the laws of exponents.
First, we rewrite each mixed number as a fraction. We have
This allows us to rewrite the expression as
We can now see that the expression contains the quotients of the same bases raised to positive integer powers. We can rewrite the expression to highlight taking the quotients of the same bases:
We now recall that for any rational number and positive integer exponents and such that , we have
Thus,
We know that raising a number to first power leaves it unchanged. We can also distribute the exponent of 2 over the fraction by recalling that for any rational number and positive integer . This allows us to rewrite the expression as follows:
We can now note that and , allowing us to rewrite this as
We can now evaluate the product
In our next example, we will evaluate an algebraic expression using the laws of exponents.
Example 5: Evaluating an Algebraic Expression Using the Laws of Exponents
If , , and , find the value of .
Answer
We first substitute the given values into the expression. This gives us
We can simplify this expression by evaluating the product inside the square as follows:
Thus,
We can now note that we are dividing exponential expressions with the same rational base. We can recall that the laws of exponents tell us that when dividing exponential expressions with the same rational base, we subtract the exponents, for any rational number and positive integers and with .
Hence,
Finally, we evaluate the exponent:
In our final example, we will simplify an algebraic expression using the laws of exponents.
Example 6: Simplifying an Algebraic Expression Using the Laws of Exponents
Given that x is a rational number, which of the following expressions is equivalent to ?
Answer
Since is a rational number, we can apply the laws of exponents to simplify this algebraic expression. We note that the expression contains the product of exponential expressions with the same rational base and the repeated exponentiation of a rational base. So, we recall the two laws of exponents for any rational number and positive integers and .
Thus, and
We can then substitute these into the expression to get which we can see is option A.
Letโs finish by recapping some of the important points from this explainer.
Key Points
- If and is a positive integer, then
- If and and are positive integers, then
- If and and are positive integers with , then
- If and and are positive integers, then
- We can use the laws of exponents to simplify exponential expressions.