In this explainer, we will learn how to perform operations and simplifications on expressions that involve integer exponents.

To help us understand how to simplify expressions involving integer exponents, we will recall how to evaluate a power. Letβs first recall how to evaluate a power with a positive integer exponent.

### Definition: Evaluating a Power

For a power with base where and exponent where , then

Next, we will recall how to evaluate a power with a zero or negative base.

### Laws: Rules for Zero and Negative Exponents

- The law for zero exponents: where
- The law for negative exponents: where and

Using this definition along with the order of operations, we can simplify numerical and algebraic expressions with integer exponents. We will do this in our first example.

### Example 1: Evaluating an Expression Involving Integer Exponents

Calculate .

### Answer

To calculate , we can write the powers in expanded form and then cancel common factors in the numerator and denominator.

Recall that the expanded form is when we write a power with base where and exponent where as

In our example, this would give us

We can cancel common factors by writing the numerator and denominator as products of prime factors first and then canceling. Doing so gives us

Therefore,

Sometimes, when simplifying expressions with integer exponents, we need to use the laws of exponents to do this. Letβs recall the laws of exponents.

### 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 ,

In our next example, we will use the rules of exponents to simplify an expression with square roots. Recall that for , . We will also use this in the next example.

### Example 2: Simplifying Numerical Expressions Involving Square Roots Using Laws of Exponents

Simplify .

### Answer

To simplify the expression , we can use laws of exponents. We will start by simplifying the numerator .

As the bases are different, it is helpful to write as and use the power of a product rule, which states where , and . For , this gives us

Substituting this back into the numerator, we get .

For the powers with base , we can then use the product rule, which states where and , . This then gives us

Since 15 is an odd exponent, then , giving us

Substituting this back into the expression, we get

Next, we can use the quotient rule for exponents, which states where , , and .

Applying this, we get

To simplify , we can write this as a product of squares and then use the rule

This then gives us

Therefore, simplified is .

Note that we can also evaluate using the power rule as follows:

In the next example, we will consider how to simplify algebraic expressions using the laws of exponents.

### Example 3: Using Laws of Exponents to Evaluate Algebraic Expressions

Simplify .

### Answer

To simplify the expression , we need to use the laws of exponents.

First, we will start by simplifying the numerator, . To do so, we use the product rule for exponents, which states that where and , . This gives us

Next, we will simplify the denominator, . In order to apply the rules for exponents, we need the bases to be the same. As such, we want to write with a base of 4. Since 16 is , then we can write

To simplify this further, we can use the power law for exponents, which states that where and , . This then gives us

Substituting back into the denominator, we now have

Since the bases are now the same, we can use the product rule for exponents again to simplify the denominator. This gives us

Substituting into the numerator and into the denominator of our original expression gives us

As we have a division, we can use the quotient rule for exponents to simplify the expression further. This rule states that where , , and . This then gives us

Evaluating , we get . Therefore,

Next, we will look at an example where the laws of exponents are used along with common algebraic methods to simplify.

### Example 4: Simplifying Rational Algebraic Expressions Using Laws of Exponents

Simplify .

### Answer

To simplify , we need to consider which laws of exponents can be used and when.

Since the numerator of the expression contains products of powers of the same base, then we can use the product rule. This rule states where and , .

Therefore, for the numerator, we get

As the denominator of the expressions contains the addition of powers with the base, then we need to be careful not to incorrectly apply the product rule. Instead, we can gather like terms by adding up the number of s, giving us 6 lots of as follows:

Since we are multiplying 6 by , then we can apply the product rule again to get

Substituting the numerator and denominator back into the expression, we get

As both the numerator and denominator have the same base, then we can apply the quotient rule for exponents. This rule states where and , .

Therefore, applying this, we get

Therefore,

In the following example, we will use the laws of exponents to simplify algebraic expressions with two different bases.

### Example 5: Simplifying Algebraic Fractions Using Properties of Exponents

Simplify .

### Answer

To simplify the expression , we can use the laws of exponents. Since the laws only apply to exponents with the same bases, then it is helpful to rewrite the expression as a product of two fractions that have numerators and denominators with the same base.

This gives us

Taking the first fraction, , we can simplify this using the quotient rule for exponents, which states where and , .

This then gives us

Next, we will use the same law to simplify the second denominator, , giving us

Substituting and back into , we get

In the last example, we will discuss how to simplify an algebraic expression using laws of exponents and by taking out common factors that are powers.

### Example 6: Using Laws of Exponents to Find an Unknown in a Given Equation

Find the value of for which .

### Answer

To find the value of in the equation , we need the left-hand side of the equation to be in the form of the right-hand side. In other words, we want it to be in the form of , meaning . To do this, we must first manipulate the powers on the left-hand side so that they are written as .

For , we can use the product rule for exponents, which states where and , . This then gives us which is in the form required.

For , we can use the product rule again, giving us which is in the form required.

Substituting and in the left-hand side of the equation, we get

Subtracting from then gives us

Comparing the coefficients of each of the terms, then we can see that .

In this explainer, we have learned how to simplify and evaluate expressions with integer exponents. Letβs recap the key points.

### Key Points

- We can simplify exponential expressions by calculating the value of powers and canceling down common factors.
- We can simplify exponential expressions using the laws of exponents, which are as follows:
- The product rule for exponents: where and ,
- The quotient rule for exponents: where , , and
- The power of a product rule: where , and
- The power of a quotient rule: where , and
- The power rule for exponents: where and ,

- We can use the laws of exponents to solve equations.