# Prerequisites

## Radicals and Rational Exponents


### Learning Objective:

1. Investigate the discipline called learning science and the idea of a knowledge space.


### Objective 1: Investigate the discipline called learning science and the idea of a knowledge space.

The brain is a complex organ. It is the control center for our bodies, while the mind is where thinking and learning take place. In an attempt to understand the processes that occur in learning, researchers study a collection of disciplines called learning sciences. This interdisciplinary field includes study of psychological, sociological, anthropological, and computational approaches to learning.

In this skill sheet we will investigate the mathematics of mastery and knowledge spaces. A knowledge space includes the possible states of knowledge of a human learner. The theory of knowledge space was introduced in 1985 by mathematical psychologists Jean-Paul Doignon and Jean-Claude Falmagne and has since been studied by many researchers.
Doignon, J.-P.; Falmagne, J.-Cl. (1985), "Spaces for the assessment of knowledge", International Journal of Man-Machine Studies.


### Practice Makes Perfect

Investigation:  There are 32 student-learning outcomes (SLO’s) in a typical College Algebra course. These are topics a student needs to master to show proficiency in College Algebra. Let’s begin by looking at just a few of these skills.  Let’s assign the variables, A, B, C, and D to the following topics. We will name the set containing each of these 4 topics, Q.


1. A = Graph the basic functions listed in the library of functions.
2. B = Find the domain of a function defined by an equation.
3. C = Create a new function through composition of functions.
4. D = Find linear functions that model data sets.
Using roster notation Q = {A, B, C, D}. 
A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in , and use the Pythagorean Theorem.
Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.

### Evaluating Square Roots

When the square root of a number is squared, the result is the original number. Since  
  the square root of  
  
 is  
  
 The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.
In general terms, if  
  
 is a positive real number, then the square root of  
  
 is a number that, when multiplied by itself, gives  
  
 The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root is the nonnegative number that when multiplied by itself equals  
  
 The square root obtained using a calculator is the principal square root.
The principal square root of  
  
 is written as  
  
 The symbol is called a radical, the term under the symbol is called the radicand, and the entire expression is called a radical expression.

### Using the Product Rule to Simplify Square Roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite  
  
 as  
  
 We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

### Using the Quotient Rule to Simplify Square Roots

Just as we can rewrite the square root of a product as a product of square roots, so too can we rewrite the square root of a quotient as a quotient of square roots, using the quotient rule for simplifying square roots. It can be helpful to separate the numerator and denominator of a fraction under a radical so that we can take their square roots separately. We can rewrite  
  
 as  
 


### Adding and Subtracting Square Roots

We can add or subtract radical expressions only when they have the same radicand and when they have the same radical type such as square roots. For example, the sum of  
  
 and  
  
 is  
  
 However, it is often possible to simplify radical expressions, and that may change the radicand. The radical expression  
  
 can be written with a  
  
 in the radicand, as  
  so  
 


### Rationalizing Denominators

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator.
We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.
For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is  
  multiply by  
 

For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is  
  then the conjugate is  
 


### Using Rational Roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.

### Understanding nth Roots

Suppose we know that  
  
 We want to find what number raised to the 3rd power is equal to 8. Since  
  we say that 2 is the cube root of 8.
The nth root of  
  
 is a number that, when raised to the nth power, gives  
  
 For example,  
  
 is the 5th root of  
  
 because  
  
 If  
  
 is a real number with at least one nth root, then the principal  of  
  
 is the number with the same sign as  
  
 that, when raised to the nth power, equals  
 

The principal nth root of  
  
 is written as  
  where  
  
 is a positive integer greater than or equal to 2. In the radical expression,  
  
 is called the index of the radical.

### Using Rational Exponents

Radical expressions can also be written without using the radical symbol. We can use rational (fractional) exponents. The index must be a positive integer. If the index  
  
 is even, then  
  
 cannot be negative.
We can also have rational exponents with numerators other than 1. In these cases, the exponent must be a fraction in lowest terms. We raise the base to a power and take an nth root. The numerator tells us the power and the denominator tells us the root.
All of the properties of exponents that we learned for integer exponents also hold for rational exponents.

### Key Concepts

1. The principal square root of a number  
  
 is the nonnegative number that when multiplied by itself equals  
  
 See .
2. If  
  
 and  
  
 are nonnegative, the square root of the product  
  
 is equal to the product of the square roots of  
  
 and  
  
 See  and .
3. If  
  
 and  
  
 are nonnegative, the square root of the quotient  
  
 is equal to the quotient of the square roots of  
  
 and  
  
 See  and .
4. We can add and subtract radical expressions if they have the same radicand and the same index. See  and .
5. Radical expressions written in simplest form do not contain a radical in the denominator. To eliminate the square root radical from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. See  and .
6. The principal nth root of  
  
 is the number with the same sign as  
  
 that when raised to the nth power equals  
  
 These roots have the same properties as square roots. See .
7. Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See  and .
8. The properties of exponents apply to rational exponents. See .

### Section Exercises


### Verbal


### Numeric

For the following exercises, simplify each expression.

### Algebraic

For the following exercises, simplify each expression.

### Real-World Applications


### Extensions

For the following exercises, simplify each expression.