# Lesson Explainer: Powers and Exponents for Rational Numbers Mathematics

In this explainer, we will learn how to identify the base and exponent in power formulas, write them in exponential and expanded forms, and evaluate simple powers.

We begin by recalling that we can represent repeated multiplication as a power. For example, is defined as the product of five twos as follows.

We call 2 the base and 5 the exponent. We can extend this definition to general rational bases. In this case, if is a positive integer and is a rational number, then will be the product of lots of .

For example, we can evaluate by finding the product of three halves as follows:

Similarly, we can evaluate by repeated multiplication as shown:

We can follow this same process in reverse. Let’s say we want to write in the exponential form. We can factor the numerator and denominator into primes as follows:

We can then split the multiplication as shown:

We see that is the product of three lots of , so we can write this in the exponential form

Thus,

Let’s now see some examples involving the powers of rational numbers.

### Example 1: Understanding Powers

What terminology do we use to describe the in the expression and the 5 in the expression ?

We recall that an expression of the form is called an exponential expression or the power of . We call the base of the expression and the exponent or power.

In the expression , we note that is the number that is being taken to a power and 5 is the power itself. Hence, is called the base of the expression and 5 is the exponent of the expression.

In our next example, we will simplify an expression by rewriting it in the exponential form.

### Example 2: Writing a Numerical Expression as an Exponent

What is ?

We could evaluate this expression by multiplying all of the numerators and denominators. This would give us

However, the options are given as powers. So, instead of evaluating these expressions, we can simplify by recalling that repeated multiplication can be rewritten as exponentiation.

In particular, the product of 7 lots of can be written by raising to an exponent of 7.

We have seen that positive integer powers can be thought of as repeated multiplication of the base. In general, if and is a positive integer, we have

We can evaluate the right-hand side of the equation by multiplying the numerators and denominators separately as follows:

In other words, we can raise a rational number to a positive integer exponent by raising its numerator and denominator to the exponent separately.

For example, we saw that . We could have instead evaluated this by cubing the numerator and denominator:

We can write this result formally as follows.

### Property: Powers of Rational Numbers

Since a positive integer power of a rational base is defined by repeated multiplication, we can show that if is a positive integer and , then

In other words, we can raise the numerator and denominator to the power separately.

In our next example, we will evaluate a power where the base is rational.

### Example 3: Evaluating Rational Numbers Raised to a Power

Find the value of , giving your answer in its simplest form.

We can evaluate this expression in two ways. We begin by recalling that exponentiation is defined by repeated multiplication. So, is the product of 3 lots of as shown:

We can multiply the numerators and denominators separately to get

We can also evaluate this expression by recalling the general result for powers of rational numbers. In general, if is a positive integer and , then

Hence,

In our next example, we will evaluate an exponential expression involving multiple exponential factors.

### Example 4: Calculating a Numerical Expression Using Powers

Evaluate the expression , giving your answer as a fraction in simplest form.

To evaluate this expression, we first need recall that the order of operations tells us to start with the powers. We then recall that if is a positive integer and , then

Hence, we can evaluate each power as follows:

We can substitute these values into the expression to get

Next, we need to evaluate the innermost parentheses. We do this by multiplying the numerators and denominators separately. We have

We can note that both 36 and 27 are divisible by 9 to help simplify.

Substituting this value into the expression gives

We can now recall that dividing by a fraction is the same as multiplying by its reciprocal. This gives

We can then simplify this expression by factoring as follows:

In our next example, we will find an expression for the volume of a cube from a given expression for its side length.

### Example 5: Solving a Word Problem by Taking Powers of Rational Numbers

Find an expression for the volume of the given cube whose side lengths are .

We begin by recalling that the volume of a cube is given by the cube of its side length. So, if the cube has side length , then its volume is . In this case, the side length is , so its volume is given by the cube of this expression: .

We can simplify this by writing the product out in full. We find the product of 3 lots of to get

We then recall that to multiply monomials, we multiply the coefficients and add the powers of the shared variables. Thus,

In our final example, we will evaluate an algebraic expression using the results for the powers of rational numbers.

### Example 6: Evaluating an Algebraic Expression Using Powers

If and , find the value of , giving your answer as a fraction in simplest form.

We first substitute the given values into the expression to get

We now want to evaluate the powers by recalling that if is a positive integer and , then

So,

We can now substitute these values into the expression to get

We can now evaluate each product by multiplying the numerators and denominators separately. We get

We can cancel the shared factor of 4 in the first term and 2 in the second term to get

We can then simplify

Finally, we need to rewrite both fractions to have the same denominator to add them together. We note that 5 is a factor of 125 , so 125 is the lowest common multiple of the denominators. Rewriting the first term to have a denominator of 125 and adding the two fractions together gives us

We note that there are no shared factors in the numerator and denominator, so we cannot simply further.

Hence,

Let’s finish by recapping some of the important points from this explainer.

### Key Points

• In an expression of the form , we call the base and the power or exponent. We can read this expression as raised to the power.
• We define positive integer powers by repeated multiplication, known as the expanded form. In general, if and is a positive integer, then is a product of lots of .
• In general, we can evaluate the power of a rational number by evaluating the power of the numerator and denominator separately. If is a positive integer and , then