Radicals and Rational Exponents

Evaluate and perform U operations with higher roots.

study Tip
Some higher even and odd roots
occur so frequently that you
might want to memorize them.

Cube Roots

Fourth
Roots

Fifth
Roots

Rationalize the denominator:

Other Kinds of Roots
We define the principal nth root of a real number a, symbolized by, as follows:

Definition of the Principal nth Root of a Real Number
means that b^n = a.
If n, the index is even, then a is nonnegative (a ≥ 0) and b is also nonnegative
(b ≥ 0). If n is odd, a and b can be any real numbers.

For example,
because 4³ = 64 and because (-2)⁵ = -32.

The same vocabulary that we learned for square roots applies to nth roots. The
symbol is called a radical and a is called the radicand.

A number that is the nth power of a rational number is called a perfect nth
power. For example, 8 is a perfect third power, or perfect cube, because 8 = 2³. Thus,
In general, one of the following rules can be used to find nth roots
of perfect nth powers:

Finding nth Roots of Perfect nth Powers
If n is odd,
If n is even,

For example,

Absolute value is not needed with odd
roofs, but is necessary with even roofs.

The Product and Quotient Rules for Other Roots
The product and quotient rule apply to cube roots, fourth roots, and all higher roots.

The Product and Quotient Rules for nth Roots
For all real number, where the indicated roots represent real numbers,

EXAMPLE· 8 Simplifying, Multiplying, and Dividing Higher Roots
Simplify:

Solution

	Find the greatest perfect cube that is a factor of 24. 23 = 8, so 8 is a perfect cube and is the greatest perfect cube factor of 24.
	Find the greatest perfect fourth power that is a factor of 32. 24 = 16. so 16 is a perfect fourth power and is the greatest perfect fourth power that is a factor of 32.

Understand and use rational exponents.

Simplify:

We have seen that adding and subtracting square roots often involves simplifying
terms. The same idea applies to adding and subtracting nth roots.

EXAMPLE 9 Combining Cube Roots
Subtract:

Solution

Factor 16. 8 is the greatest perfect cube factor: Multiply: 5·2 = 10. Applythe distributive property. Simplify.

Subtract:

Rational Exponents
We define rational exponents so that their properties are the same as the properties
for integer exponents. For example, we know that exponents are multiplied when an
exponential expression is raised to a power. For this to be true,

We also know that

Can you see that the square of both and is 7? It is reasonable to conclude
that

means
We can generalize the fact that means with the following definition:

The Definition of
If represents a real number and n ≥ 2 is an integer, then

The denominator of the rational
exponent is the radical's index.

Furthermore,

EXAMPLE 10 Using the Definition of
Simplify:

Solution

The denominator is the index.

The base is 16 and the negative sign is not affected by the exponent.

Parentheses show that the base is -27 and that the negative sign is affected by the exponent.

Simplify:

In Example 10 and Check Point 10 each rational exponent had a numerator of
1. If the numerator is some other integer, we till want to multiply exponent when
raising a power to a power. For this reason,

Thus,

Do you see that the denominator, 3, of the rational exponent is the same as the
index of the radical? The numerator, 2, of the rational exponent serves as an
exponent in each of the two radical forms. We generalize these ideas with the following
definition:

The Definition of
If represents a real number and m/n is a positive rational number, n≥2, then

Also,

Furthermore, if is a nonzero real number, then

The first form of the definition of shown again below, involves taking the root
first. This form is often preferable because smaller numbers are involved. Notice that
the rational exponent consists of two parts, indicated by the following voice balloons:
The numerator is the exponent.

EXAMPLE 11 Using the Definition of
Simplify:

Solution

Technology
Here are the calculator keystroke
sequences for

Many Scientific Calculators

Many Graphing Calculators

Simplify:

Properties of exponents can be applied to expressions containing rational
exponents.

EXAMPLE 12 Simplifying Expressions with Rational Exponents
Simplify using properties of exponents:

Solution

	Group factors with the same base. When multiplying expressions with the same base, add the exponents.
	Group factors with the same base. When dividing expressions with the same base, subtract the exponents.

Simplify: a
Rational exponents are sometimes useful for simplifying radicals by reducing
their index.

EXAMPLE 13 Reducing the Index of a Radical
Simplify:
Solution

Simplify:

EXERCISE SET P.3

Practice Exercises
Evaluate each expression in Exercises 1-12, or indicate
that the root is not a real number.

Use the product rule to simplify the expressions in Exercises
13-22. In Exercises 17-22, assume that variable represent nonnegative
real numbers.

Use the quotient rule to simplify the expressions in Exercises
23-32. Assume that x > 0.


In Exercises 33-44, add or subtract terms whenever possible.

In Exercises 45-54, rationalize the denominator.

Evaluate each expression in Exercises 55--66, or indicate that the
root is not a real number.

Simplify (he radical expressions in Exercises 67-74.

In Exercises 75-82, add or subtract terms whenever possible.

In Exercises 83-90, evaluate each expression without using a
calculator.

In Exercises 91-100, simplify using properties of exponents.

1n Exercises 101-108, simplify by reducing (he index of the radical.

Practice Plus
In Exercises 109-110, evaluate each expression.

In Exercises 111-114, simplify each expression. Assume that all
variables represent positive numbers.

Application Exercises
The formula

models the distance, d, in miles, that a person h feet high can see to
the horizon. Use this formula to solve Exercises 115- 116.

115. The pool deck on a cruise ship is 72 feet above the water.
How far can passengers on the pool deck see? Write the
answer in simplified radical form. Then use the simplified
radical form and a calculator to express the answer to the
nearest tenth of a mile.

116. The captain of a cruise ship is on the star deck, which is 120
feet above the water. How far can the captain see? Write
the answer in simplified radical form. Then u e the simplified
radical form and a calculator to express the answer to
the nearest tenth of a mile.

Police use the formula to estimate the speed of a car, v,
in miles per hour, based on the length, L, in feet, of its skid marks
upon sudden braking on a dry asphalt road. Use the formula to
solve Exercises 117-118.

117. A motorist is involved in an accident. A police officer measures
the car's skid marks to be 245 feet long. Estimate the
peed at which the motorist was traveling before braking. If
the posted speed limit is 50 miles per hour and the motorist
tells the officer he was not speeding, should the officer
believe him? Explain.

118. A motorist is involved in an accident. A police officer measures
the car's skid marks to be 45 feet long. Estimate the
speed at which the motorist was traveling before braking. If
the posted speed limit is 35 miles per hour and the motorist
tells the officer she was not speeding, should the officer
believe her? Explain.

119.In the Peanuts cartoon shown below, Woodstock appears to
be working step mentally. Fill in the mi sing steps that show
how to go from

PEA NUTS reprinted by permission of United Feature
Syndicate, Inc.