Rational Numbers and Exponents

RATIONAL NUMBERS AND EXPONENTS

Unit Overview
In this unit, you will learn about rational numbers. You will learn how to simplify rational expressions and solve equations that contain rational numbers. This unit will also introduce you to the properties of exponents so you will be able to simplify polynomials in a later unit.

Rational Numbers

A rational number is any number that can be expressed as a ratio of two integers,

, where b ≠ 0.

Example #1:

4 is a rational number because it can be written as

Example #2:

–8.3 is a rational number because it can be written as

Therefore, in general, a rational number may be written in the form of a fraction or a decimal.

Review: In this course, we will not explain how to add, subtract, multiply, or divide rational numbers extensively as you should have learned these processes in earlier courses. Therefore, we will only go through a couple of examples to refresh your memory.

Adding/Subtracting:

*Must have a common denominator, and then add or subtract the numerators.

Adding and Subtracting Fractions--Running and Driving (03:36)

Multiplying:

*Cancel any common factors and multiply straight across.

Multiplying Fractions--Vacation Time (01:37)

Dividing:

*Multiply by the reciprocal.

Dividing Fractions--Baking Bread (02:11)

Stop! Go to Questions #1-8 about this section, then return to continue on to the next section.

Solving Equations with Rational Numbers

When solving equations involving rational numbers, you will use the same principle of opposite operations as you did when solving equations with integers. One exception is that when dividing by a rational fraction, remember to multiply by the reciprocal.

Let’s try a few examples:

Example #1: Solve

for x.

Example #2: Solve y + 6.8 = 2.5 for y.

Example #3:

Example #4:

Example #5:

Stop! Go to Questions #9-15 about this section, then return to continue on to the next section.

Exponent Properties

Recall that exponents tell us how many times we use the base as a factor. For example, 3⁴ = 3 ⋅ 3 ⋅ 3 ⋅ 3. An expression written with exponents is said to be in exponential form.

When simplifying algebraic expressions involving exponents, there are a few rules (properties) apply when completely simplifying the expression. Those properties are listed below with examples.

Example #1: Simplify x⁷ ⋅ x³.

x⁷ ⋅ x³ = x^{7 + 3} = x¹⁰*Add the exponents.

Example #2: Simplify z⁵ ⋅ z² · z.

z⁵ ⋅ z² ⋅ z = z⁵ ⋅ z² ⋅ z¹ = z^{(5 + 2 + 1)} = z⁸

Example #3: Simplify

= x^{4 × 3} = x¹² *Multiply the exponents.

Example #4: Simplify (2x)³.

(2x)³ = 2³ ⋅ x³= 8x³ *Raise all bases to the power.

Example #5: Simplify (−5m³n²)⁴.

(−5m³n²)⁴ = (−5)⁴ ⋅ m^{3 × 4} ⋅ n^{2 × 4} = 625m¹²n⁸

Example #6: Simplify

*Raise numerator and denominator to the power.

Example #7: Simplify

Example #8: Simplify

*Subtract the exponents.

Example #9: Simplify

*Exponential expressions are not considered completely simplified unless the exponents are positive; for this, the next property is especially important.

Example #10: Simplify x⁻³.

x⁻³ =

*The reciprocal of x is

Example #11: Simplify

= x³*The reciprocal of

is x.

Example #12: Simplify

*The reciprocal of is x.

Below is a link to Practice with Exponents, a great website that gives you an opportunity to practice some problems. You can check your answer immediately.

Start with problems labeled Easier. You will get a chance to try the harder problems a little later in this unit.
Use scrap paper as you work out the expressions.
Try numerous problems, but only spend about 15 minutes doing so.
Make sure you understand the properties of exponents and how they are used so that you will be able to complete the problems in the questions area.

Click here to begin the practice.

Stop! Go to Questions #16-22 about this section, then return to continue on to the next section.

Multiplying and Dividing Monomials

In the previous section, each of the exponent properties used a term called a monomial. A monomial can be a number, a variable, or products of numbers and variables. For example, 3, x, and 3xy are all considered monomials.

When multiplying or dividing monomials, the properties of exponents must be followed as shown in the examples below:

Example #1: Simplify (3x²)(4x⁶).

Example #2: Simplify

We will use this concept in future units when we multiply monomials with polynomials.

Below is a link to Practice with Exponents, a great website that gives you an opportunity to practice some problems. You can check your answer immediately.

Start with problems labeled Medium.
Move to more difficult problems as you answer questions correctly.
Use scrap paper as you work out the expressions.
Try numerous problems, but only spend about 15 minutes doing so.
Make sure you understand the properties of exponents and how they are used so that you will be able to complete the problems in the questions area.

Click here to begin the practice.

Stop! Go to Questions #23-24 about this section, then return to continue on to the next section.

Rational Exponents

Did you know that there is more than one way to write a root?

An alternate way to express roots is to use rational (fractional) exponents and can be very useful when dealing with more complicated expressions.

First, let's review the terms associated with radicals.

Now, let’s examine the rational exponent rule.

This rule can be interpreted as the nth root of m raised to the power of b equals the m (the base) raised to the

power.

As you can see, when the exponent is a rational (fractional) number, this is actually a root and vice versa.

Let’s look at some examples below.

What is the rational exponent expression for ?

x raised to the 1/2 power

"Click here" to check the answer.

In the examples above, the b is a one as described in the rule

.

Let’s examine some examples when b is not a one.

What is the rational exponent expression for ?

x raised to the 3/5 power

"Click here" to check the answer.

Now, let’s evaluate using our new knowledge of rational exponents.

Example 1:

Example 2:

Example 3: Simplify the expression

Now, apply the rational exponent rule.

What is the rational exponent expression for ?

The cube root of 8 to the 5th power = 2 × 2 × 2 × 2 × 2 = 32

"Click here" to check the answer.

Strategies for Simplifying Other Roots and Rational Exponents (04:49)

Now, let’s apply some of the rational exponents using the property of exponents.

Example 4: Simplify the expression