College Algebra Fifth Edition James Stewart Lothar Redlin Saleem Watson

Rational Exponents and Radicals P.5

Exponents and Radicals In this section, we learn to work with expressions that contain radicals or rational exponents.

Radicals

We know what 2 n means whenever n is an integer. To give meaning to a power, such as 2 4/5, whose exponent is a rational number, we need to discuss radicals.

Radicals The symbol √ means: “the positive square root of.” Thus,

Radicals Since a = b 2 ≥ 0, the symbol makes sense only when a ≥ 0. For instance,

nth Root Square roots are special cases of nth roots. The nth root of x is the number that, when raised to the nth power, gives x.

nth Root—Definition If n is any positive integer, then the principal nth root of a is defined as follows: If n is even, we must have a ≥ 0 and b ≥ 0.

nth Roots Thus,

nth Roots However,,, and are not defined. For instance, is not defined because the square of every real number is nonnegative.

nth Roots Notice that So, the equation is not always true. It is true only when a ≥ 0.

nth Roots However, we can always write This last equation is true not only for square roots, but for any even root. This and other rules used in working with nth roots are listed in the following box.

Properties of nth Roots In each property, we assume that all the given roots exist.

E.g. 1—Simplifying Expressions Involving nth Roots Simplify these expressions.

E.g. 8—Expressions with nth Roots Example (a)

E.g. 8—Expressions with nth Roots Example (b)

Combining Radicals It is frequently useful to combine like radicals in an expression such as This can be done using the Distributive Property. Thus, The next example further illustrates this process.

E.g. 2—Combining Radicals

Rational Exponents

To define what is meant by a rational exponent or, equivalently, a fractional exponent such as a 1/3, we need to use radicals.

Rational Exponents To give meaning to the symbol a 1/n in a way that is consistent with the Laws of Exponents, we would have to have: (a 1/n ) n = a (1/n)n = a 1 = a So, by the definition of nth root, In general, we define rational exponents as follows.

Rational Exponent—Definition For any rational exponent m/n in lowest terms, where m and n are integers and n > 0, we define If n is even, we require that a ≥ 0. With this definition, it can be proved that the Laws of Exponents also hold for rational exponents.

E.g. 3—Using the Definition of Rational Exponents

E.g. 4—Using the Laws of Exponents

E.g. 5—Writing Radicals as Rational Exponents

Rationalizing the Denominator

It is often useful to eliminate the radical in a denominator by multiplying both numerator and denominator by an appropriate expression. This procedure is called rationalizing the denominator.

Rationalizing the Denominator If the denominator is of the form, we multiply numerator and denominator by. In doing this, we multiply the given quantity by 1. So, we do not change its value.

Rationalizing the Denominator For instance, Note that the denominator in the last fraction contains no radical.

Rationalizing the Denominator In general, if the denominator is of the form with m < n then multiplying the numerator and denominator by will rationalize the denominator. This is because (for a > 0)

E.g. 6—Rationalizing Denominators