In this explainer, we will learn how to apply the laws of exponents to multiply and divide powers and to calculate a power raised to a power over the real numbers.
Recall that to evaluate a power, we multiply the base by the exponent the number of times indicated. Let’s recap the definition below.
Definition: Evaluating a Power
For a power with base , , and exponent , , then
By using our understanding of how to evaluate power, we can derive the rules of how to multiply and divide powers as well as the laws for a power raised to a power.
Let’s say we have two powers with base , , where the first is raised to the power of and the second is raised to the power of , , . If we multiply the two powers together, we get
If we count the number of times is multiplied by itself, this is times plus times, which is a total of times. This then gives us which is the product rule for exponents.
Now, let’s say we have two powers with base , , where the first is raised to the power of and the second is raised to the power of , , . If we divide the first power by the second power, we get
If we cancel common factors in the numerator and the denominator, we end up canceling all the ’s in the numerator, meaning we have lots of minus lots of , leaving us with lots of . In other words, we have
This then gives us, when .
If we consider the case when is larger than , so , then we can use the same approach, but with a higher power in the denominator, giving us
Canceling common factors, we get
If we then use the law for negative indices, we get
Therefore, for all integer values of and , which is the quotient rule for exponents.
Next, let’s say we have a power with a base of , , , which is raised to the power of , . This gives us
If we rewrite this so that all of the ’s and all of the ’s are together, we get
Since the number of times is multiplied by itself is times and similarly the number of times is multiplied by itself is times, then we get which is the power of a product rule for exponents.
Similarly, let’s say we have a power with a base of , , , which is raised to the power of , . This gives us
If we rewrite this as one fraction, with the ’s in the numerator and the ’s in the denominator, we get
Since the number of times is multiplied by itself is times and similarly the number of times is multiplied by itself is times, then we get which is the power of a quotient rule for exponents.
Lastly, if we have now only one power with base , , which is raised to the power of and then raised to another power , , , we get the following:
If we count the number of times is multiplied by itself, this is times multiplied by times, which is a total of times. This then gives us which is called the power rule for exponents.
Let’s summarize the laws in the rules below.
Rule: Laws of Exponents
The following are the rules of exponents with real bases and integer exponents:
- The product rule: where and ,.
- The quotient rule: where , , and .
- The power of a product rule: where , and .
- The power of a quotient rule: where , , and .
- The power rule: where and , .
As we are working with bases with real numbers, then often we need to simplify expressions with square roots. As such, it is helpful to recall the following definition.
Definition: Simplifying Expressions with Squares and Square Roots
For an expression with a base , where ,
- ;
- .
In our first example, we will discuss how to use the product rule for exponents to simplify an expression with a real base.
Example 1: Simplifying an Expression Involving Multiplication of Positive Integer Powers in the Real Numbers
Fill in the blank: The expression simplifies to .
Answer
To find what is equal to, we need to simplify the expression. As we have powers with the same base being multiplied together, then we can use the product rule for exponents, which states that where and , .
Starting with , we can write this as . Then, applying the product rule, we get
Substituting back into the original expression, we get
Again, we can apply the product rule, which gives us
Lastly, to simplify , we can use the property , where . To do so, we can write as a product of powers of 2 using the product law in reverse. This gives us
Therefore, simplifies to 27.
In the next example, we will use the quotient rule for exponents to simplify an expression with a real base.
Example 2: Simplifying an Expression Involving Division of Positive Integer Powers in the Real Numbers
Which of the following is equal to ?
Answer
To find what is equal to, we need to simplify the expression. As we have two powers with the same base, where one is divided by the other, then we can use the quotient rule for exponents, which states that where , , and .
Applying this rule with , , and , we get
Comparing with the options given, we can see that this is the same as option D, which is the correct answer.
In the next example, we will consider how to use rules for exponents to simplify expressions with square roots in them, where one base is the negative of another.
Example 3: Simplifying an Expression Involving Division of Positive Integer Powers in the Real Numbers
What is the value of ?
Answer
To find the value of , we need to first simplify the expression. One approach is to use the laws of exponents.
We can see that the bases are nearly the same for the two parts of the expression, except that the first base is the negative of the second base. Instead, we can write the expression as
Using the power of a product rule for exponents, which states that where , and , we can apply this to , giving us
Since and now have the same base, we can apply the quotient rule for exponents, which states that where and , , giving us
We can now evaluate this, by calculating and using and the product rule for exponents to find , as follows:
Therefore, the value of is .
As with mathematical operations in general, we can apply multiple operations in one expression and use the laws of exponents to simplify these. We usually apply the laws according to the order of operations. We will explore this further in the next example.
Example 4: Evaluating a Numerical Expression Involving Real Numbers Using the Law of Exponents
Simplify .
Answer
To simplify , we need to use the laws of exponents. To do so, it is helpful to apply the laws to the numerator first, which is .
To simplify , we can use the product rule for exponents, which states that where , , , giving us
Substituting this back into the expression, we get . To simplify further, we can use the quotient rule for exponents, which states that where , , and , giving us
We can use the rule and the product rule to simplify this further, giving us
Therefore, simplified gives us 25.
Next, we will consider how to simplify expressions where there are negative exponents. Let’s recall the laws of zero and negative exponents.
Laws: Rules for Zero and Negative Exponents
- The law for zero exponents: where .
- The law for negative exponents: where and .
Using the law for negative exponents, we can derive two other properties.
First, if we take , where and , and multiply both sides by , we get
Second, if we apply the law for negative exponents to a power with base , where , and exponent , we get
By the power of a quotient rule, which states that where , , and , we then get
If we multiply the numerator and denominator by , we then get
If we then apply the power of a quotient rule again, we get
Similarly, we can show that
Let’s summarize these two rules.
Laws: Rules for Zero and Negative Exponents
- The law of multiplicative inverses: where and .
- The power of a quotient rule for negative exponents: where , and .
In the next example, we will use the law for negative exponents and the laws of indices to evaluate an expression.
Example 5: Evaluating a Numerical Expression Involving Real Numbers Using the Law of Exponents
For and , evaluate .
Answer
To evaluate , we must start by substituting and . This gives us
Next, we can use the power of a quotient rule for negative exponents to simplify , which states that where , and , giving us
Substituting this back to the original expression, we have
Using the power of a quotient rule, which states that where , , and , we get and
This then gives us
If we expand the powers, we get
Canceling common factors then gives us
Therefore, for and , the value of is .
So far, we have considered how to use the laws of exponents to simplify expressions with real bases. Next, we will consider how to solve simple exponential equations using the laws of exponents. Recall that if two powers are equal, then if either the bases or the exponents are also equal, the following rules apply.
Rule: Equating Powers with the Same Base or the Same Exponent
- If , where , then .
- If , then when and when .
In the next example, we will discuss how to use the laws of exponents to solve simple exponential equations.
Example 6: Using the Laws of Exponents to Solve a Simple Exponential Equation
Find the value of in the equation .
Answer
To find the value of in the equation , we can use the laws of exponents to simplify the different parts of the equation and then use the rule that states that if , where , then , to find the value of .
In order to apply the rules of equal powers, we need to have two powers with either the same base or the same exponent. As here the unknown is in the exponents, we need to write each side of the equation as a power with the same base. Meaning, we need to write 16 as , giving us which we can simplify using the product rule, which states that where and , .
This then gives us on the left-hand side
Looking at the right-hand side, , we can simplify this using the power rule, which states that where and , .
This then gives us on the right-hand side
Equating both sides, we get
Since the bases are both equal, then the exponents must also be equal, giving us
Solving for , we then get
Therefore, the value of is 5 in the equation .
In this explainer, we have learned how to use the laws of exponents to simplify expressions with real numbers raised to integer powers. We have also learned how to solve simple exponential equations using the laws of powers. Let’s recap the key points.
Key Points
- We can use the laws of exponents to simplify powers with real bases and integer
powers. These rules state the following:
- The product rule: where and , .
- The quotient rule: where , , and .
- The power of a product rule: where , and .
- The power of a quotient rule: where , , and .
- The power rule: where and , .
- The law of multiplicative inverses: where and .
- The power of a quotient rule for negative exponents: where , and .
- To solve simple exponential equations with real bases, we can use the following
rules when the bases or exponents are equal:
- If , where , then .
- If , then when and when .