Radicals and Rational Exponents
Evaluate and perform U operations with higher roots.
Some higher even and odd roots
occur so frequently that you
might want to memorize them.
Rationalize the denominator:
Other Kinds of Roots
We define the principal nth root of a real number a, symbolized by,
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
(b ≥ 0). If n is odd, a and b can be any real numbers.
because 43 = 64 and
because (-2)5 = -32.
The same vocabulary that we learned for square roots
applies to nth roots. The
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,
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
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
and use rational exponents.
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
||Factor 16. 8 is the greatest perfect cube factor:
Multiply: 5·2 = 10.
Applythe distributive property.
We define rational exponents so that their properties are the same as the
for integer exponents. For example, we know that exponents are multiplied when
exponential expression is raised to a power. For this to be true,
We also know that
Can you see that the square of both
is 7? It is reasonable to conclude
We can generalize the fact that
with the following definition:
EXAMPLE 10 Using the Definition of
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
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
raising a power to a power. For this reason,
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
The Definition of
represents a real number and m/n is a positive rational number, n≥2, then
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.
the rational exponent consists of two parts, indicated by the following voice
The numerator is the exponent.
EXAMPLE 11 Using the Definition of
Here are the calculator keystroke
Many Scientific Calculators
Many Graphing Calculators
Properties of exponents can be applied to expressions
EXAMPLE 12 Simplifying Expressions with Rational
Simplify using properties of exponents:
||Group factors with the same base.
When multiplying expressions with the same base, add
||Group factors with the same base.
When dividing expressions with the same base, subtract the
Rational exponents are sometimes useful for simplifying radicals by reducing
EXAMPLE 13 Reducing the Index of a Radical
EXERCISE SET P.3
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
13-22. In Exercises 17-22, assume that variable represent nonnegative
Use the quotient rule to simplify the expressions in
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
root is not a real number.
Simplify (he radical expressions in Exercises 67-74.
In Exercises 75-82, add or subtract terms whenever
In Exercises 83-90, evaluate each expression without using
In Exercises 91-100, simplify using properties of
1n Exercises 101-108, simplify by reducing (he index of
In Exercises 109-110, evaluate each expression.
In Exercises 111-114, simplify each expression. Assume
variables represent positive numbers.
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
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
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
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
be working step mentally. Fill in the mi sing steps that show
how to go from
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