Radicals and Rational Exponents – Pre-Calculus (2024)

Learning Objectives

In this section students will:

• Evaluate square roots.
• Use the product rule to simplify square roots.
• Use the quotient rule to simplify square roots.
• Add and subtract square roots.
• Rationalize denominators.
• Use rational roots.

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.

Key Concepts

• The principal square root of a number is the nonnegative number that when multiplied by itself equals See (Figure).
• If and are nonnegative, the square root of the product is equal to the product of the square roots of and See (Figure) and (Figure).
• If and are nonnegative, the square root of the quotient is equal to the quotient of the square roots of and See (Figure) and (Figure).
• We can add and subtract radical expressions if they have the same radicand and the same index. See (Figure) and (Figure).
• 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 (Figure) and (Figure).
• 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 (Figure).
• Radicals can be rewritten as rational exponents and rational exponents can be rewritten as radicals. See (Figure) and (Figure).
• The properties of exponents apply to rational exponents. See (Figure).

Section Exercises

Glossary

index

the number above the radical sign indicating the nth root

principal nth root

the number with the same sign as that when raised to the nth power equals

principal square root

the nonnegative square root of a number that, when multiplied by itself, equals

radical

the symbol used to indicate a root

radical expression

an expression containing a radical symbol

radicand

the number under the radical symbol