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 .