Precalculus Mathematics for Calculus Fifth Edition James Stewart  Lothar Redlin  Saleem Watson

Exponents and Radicals 1.2

Exponents and Radicals In this section, we give meaning to expressions such as am/n in which the exponent m/n is a rational number. To do this, we need to recall some facts about integer exponents, radicals, and nth roots.

Integer Exponents

Integer Exponents A product of identical numbers is usually written in exponential notation. For example, 5 · 5 · 5 is written as 53. In general, we have the following definition.

Exponential Notation If a is any real number and n is a positive integer, then the nth power of a is: an = a · a · · · · · a The number a is called the base and n is called the exponent.

E.g. 1—Exponential Notation

Rules for Working with Exponential Notation We can state several useful rules for working with exponential notation.

Rule for Multiplication To discover the rule for multiplication, we multiply 54 by 52: It appears that, to multiply two powers of the same base, we add their exponents.

Rule for Multiplication In general, for any real number a and any positive integers m and n, we have: Therefore, aman = am+n.

Rule for Multiplication We would like this rule to be true even when m and n are 0 or negative integers. For instance, we must have: 20 · 23 = 20+3 = 23 However, this can happen only if 20 = 1.

Rule for Multiplication Likewise, we want to have: 54 · 5–4 = 54+(–4) = 54–4 = 50 = 1 This will be true if 5–4 = 1/54. These observations lead to the following definition.

Zero and Negative Exponents If a ≠ 0 is any real number and n is a positive integer, then a0 = 1 and a–n = 1/an

E.g. 2—Zero and Negative Exponents

Laws of Exponents Familiarity with these rules is essential for our work with exponents and bases. The bases a and b are real numbers. The exponents m and n are integers.

If m and n are positive integers, we have: Law 3—Proof If m and n are positive integers, we have: The cases for which m ≤ 0 or n ≤ 0 can be proved using the definition of negative exponents.

If n is a positive integer, we have: Law 4—Proof If n is a positive integer, we have: We have used the Commutative and Associative Properties repeatedly. If n ≤ 0, Law 4 can be proved using the definition of negative exponents. You are asked to prove Laws 2 and 5 in Exercise 88.

E.g. 3—Using Laws of Exponents

E.g. 3—Using Laws of Exponents

E.g. 4—Simplifying Expressions with Exponents

E.g. 4—Simplifying Example (a)

E.g. 4—Simplifying Example (b)

Simplifying Expressions with Exponents When simplifying an expression, you will find that many different methods will lead to the same result. You should feel free to use any of the rules of exponents to arrive at your own method.

Laws of Exponents We now give two additional laws that are useful in simplifying expressions with negative exponents.

Law 7—Proof Using the definition of negative exponents and then Property 2 of fractions (Section 1-1), we have: You are asked to prove Law 6 in Exercise 88.

E.g. 5—Simplifying Exprns. with Negative Exponents Eliminate negative exponents and simplify each expression.

E.g. 5—Negative Exponents Example (a) We use Law 7, which allows us to move a number raised to a power from the numerator to the denominator (or vice versa) by changing the sign of the exponent.

E.g. 5—Negative Exponents Example (b) We use Law 6, which allows us to change the sign of the exponent of a fraction by inverting the fraction.

Scientific Notation

Scientific Notation Exponential notation is used by scientists as a compact way of writing very large numbers and very small numbers. For example, The nearest star beyond the sun, Proxima Centauri, is approximately 40,000,000,000,000 km away. The mass of a hydrogen atom is about 0.00000000000000000000000166 g.

Such numbers are difficult to read and to write. Scientific Notation Such numbers are difficult to read and to write. So, scientists usually express them in scientific notation.

Scientific Notation A positive number x is said to be written in scientific notation if it is expressed as follows: x = a x 10n where: 1 ≤ a ≤ 10. n is an integer.

Scientific Notation For instance, when we state that the distance to Proxima Centauri is 4 x 1013 km, the positive exponent 13 indicates that the decimal point should be moved 13 places to the right: 4 x 1013 = 40,000,000,000,000

Scientific Notation When we state that the mass of a hydrogen atom is 1.66 x 10–24 g, the exponent –24 indicates that the decimal point should be moved 24 places to the left: 1.66 x 10–24 = 0.00000000000000000000000166

E.g. 6—Writing Numbers in Scientific Notation

Scientific Notation in Calculators Scientific notation is often used on a calculator to display a very large or very small number. Suppose we use a calculator to square the number 1,111,111.

Scientific Notation in Calculators The display panel may show (depending on the calculator model) the approximation 1.234568 12 or 1.23468 E12 The final digits indicate the power of 10, and we interpret the result as 1.234568 x 1012.

E.g. 7—Calculating with Scientific Notation If a ≈ 0.00046 b ≈ 1.697 x 1022 and c ≈ 2.91 x 10–18 use a calculator to approximate the quotient ab/c. We could enter the data using scientific notation, or we could use laws of exponents as follows.

E.g. 7—Calculating with Scientific Notation We state the answer correct to two significant figures because the least accurate of the given numbers is stated to two significant figures.

Radicals

We know what 2n means whenever n is an integer. Radicals We know what 2n means whenever n is an integer. To give meaning to a power, such as 24/5, whose exponent is a rational number, we need to discuss radicals.

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

Since a = b2 ≥ 0, the symbol makes sense only when a ≥ 0. Radicals Since a = b2 ≥ 0, the symbol makes sense only when a ≥ 0. For instance,

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.

If n is even, we must have a ≥ 0 and b ≥ 0. 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,

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

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

However, we can always write 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. 8—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. 9—Combining Radicals Combine:

E.g. 9—Combining Radicals Example (a)

E.g. 9—Combining Radicals Example (b) If b > 0, then

Rational Exponents

Rational Exponents To define what is meant by a rational exponent or, equivalently, a fractional exponent such as a1/3, we need to use radicals.

Rational Exponents To give meaning to the symbol a1/n in a way that is consistent with the Laws of Exponents, we would have to have: (a1/n)n = a(1/n)n = a1 = 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. 10—Using the Definition of Rational Exponents

E.g. 10—Using the Definition of Rational Exponents

E.g. 11—Using the Laws of Exponents

E.g. 11—Using the Laws of Exponents

E.g. 11—Using the Laws of Exponents

E.g. 12—Writing Radicals as Rational Exponents

E.g. 12—Writing Radicals as Rational Exponents

Rationalizing the Denominator

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. 13—Rationalizing Denominators