5.1 Radical Notation and Rational Exponents


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1 Section 5.1 Radical Notation and Rational Exponents Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots (such as square roots and cube roots). Along the way, we ll define higher roots and develop a few of their properties. Integer Exponents Recall that use of a positive integer exponent is simply a shorthand for repeated multiplication. For example, 2 3 = (5.1) In general, b n stands for the quanitity b multiplied by itself n times. With this definition, the following Laws of Exponents hold. Laws of Exponents 1. b r b s = b r+s 2. b r b s = br s 3. (b r ) s = b rs Recall that negative exponents, as well as the 0 exponent, are simply defined in such a way that the Laws of Exponents will work for all integer exponents. Negative exponents and the 0 exponent are defined as follows: Definition 1 provided that b 0. b n = 1 b n and b 0 = 1 For example, 2 0 = 1, and 2 4 = 1/2 4 = 1/16. We now have b n defined for all integers n, in such a way that the Laws of Exponents hold. Recall that we can likewise define expressions using rational exponents, such as 2 1/3, in a consistent manner. Before doing reviewing that material, we ll need to take a detour and review roots. Roots Square Roots: Let s begin by reminding ourselves what the square root of a real number represents.
2 2 Chapter 5 Definition 2 Given a real number a, the square root of a is a number x such that x 2 = a. We investigate square roots in more detail by looking for solutions of the equation x 2 = a. (5.2) There are three cases, each depending on the value and sign of a. Summary: Square Roots The solutions of x 2 = a are called square roots of a. Case I: a < 0. The equation x 2 = a has no real solutions. Case II: a = 0. The equation x 2 = a has one real solution, namely x = 0. Thus, 0 = 0. Case III: a > 0. The equation x 2 = a has two real solutions, x = ± a. The notation a calls for the positive square root of a, that is, the positive solution of x 2 = a. The notation a calls for the negative square root of a, that is, the negative solution of x 2 = a. Let s look at some examples. Example 1 What are the solutions of x 2 = 5? We re asked to find a solution of x 2 = 5, so you must find a number whose square equals 5. However, whenever you square a real number, the result is always nonnegative (zero or positive). It is not possible to square a real number and get 5. That is, 5 is not a real number. Example 2 What are the solutions of x 2 = 0? There is only one solution, namely x = 0. Note that this means that 0 = 0. Example 3 What are the solutions of x 2 = 25?
3 Section 5.1 Radical Notation and Rational Exponents 3 The solutions of x 2 = 25 are called square roots of 25 and are written x = ± 25. In this case, we can simplify further and write x = ±5. Cube Roots: Let s move on to the definition of cube roots. Definition 3 Given a real number a, a cube root of a is a number x such that x 3 = a. For example, 2 is a cube root of 8 since 2 3 = 8. Likewise, 4 is a cube root of 64 since ( 4) 3 = 64. Thus, taking the cube root is the opposite of cubing, so the definition of cube root must be closely connected to the graph of y = x 3, the cubing function. Therefore, we look for solutions of A detailed summary of cube roots follows. x 3 = a. (5.3) Summary: Cube Roots The solutions of x 3 = a are called the cube roots of a. Whether a is negative, zero, or positive makes no difference. There is exactly one real solution, namely x = 3 a. Let s look at an example. Example 4 What are the solutions of x 3 = 8? The equation x 3 = 8 has exactly one real solution, denoted x = 3 8. Now since ( 2) 3 = 8, it follows that x = 2 is a real solution of x 3 = 8. Consequently, the cube root of 8 is 2, and we write 3 8 = 2. Again, because there is only one real solution of x 3 = 8, the notation 3 8 is pronounced the cube root of 8. Note that, unlike the square root of a negative number, the cube root of a negative number is allowed.
4 4 Chapter 5 Higher Roots: The previous discussions generalize easily to higher roots, such as fourth roots, fifth roots, sixth roots, etc. The key is to remember that all even roots behave the same as the square root and all odd roots behave the same as the cube root. Summary: Even nth Roots If n is a positive even integer, then the solutions of x n = a are called nth roots of a. Case I: a < 0. The equation x n = a has no real solutions. Case II: a = 0. The equation x n = a has exactly one real solution, namely x = 0. Thus, n 0 = 0. Case III: a > 0. The equation x n = a has two real solutions, x = ± n a. The notation n a calls for the positive nth root of a, that is, the positive solution of x n = a. The notation n a calls for the negative nth root of a, that is, the negative solution of x n = a. Summary: Odd nth Roots If n is a positive odd integer, then the solutions of x n = a are called the nth roots of a. Whether a is negative, zero, or positive makes no difference. There is exactly one real solution of x n = a, denoted x = n a. Rational Exponents Recall that rational exponents are defined in such a way that the Laws of Exponents still apply. For example, if n is any odd positive integer, then 2 1/n must be defined by the formula 2 1 n = n 2. With this definition, the Laws of Exponents hold for all rational exponents. Definition 4 For a positive rational exponent m n, and b > 0, For a negative rational exponent m n, b m n = n b m = ( n b) m. (5.4) b m 1 n =. (5.5) b m n For b < 0, the same definitions make sense only when n is odd. For example ( 2) 1 4 is not defined.
5 Section 5.1 Radical Notation and Rational Exponents 5 Remark 1. example ( 2) 1 4 Example 5 For b < 0, the same definitions make sense only when n is odd. For is not defined. Simplify the following expressions, and write them in the form x r : a) x x 4 = x = x = x 12 a) x 2 3 x 1 4, b) x 2 3 x 1 4, c) ( ) x b) x 2 3 x 1 4 c) ( x 2 3 = x = x = x 5 12 ) 1 4 = x = x 2 12 = x 1 6 Multiplication Properties of Radicals Recall the following property of radicals. Property 1 Let a and b be positive real numbers. Then, ab = a b. (5.6) This result can be used in two distinctly different ways. You can use the result to multiply two square roots, as in 7 5 = 35. You can also use the result to factor, as in 35 = 5 7. One question that arises is when is a radical in reduced form? We use the following set of guidelines for a special form of the answer which we will call simple radical form. The First Guideline for Simple Radical Form. When possible, factor out a perfect square. For example, 32 is not in simple radical form, as it is possible to factor out a perfect square, as in 32 = 16 2 = 4 2.
6 6 Chapter 5 Let s place another radical expression in simple radical form. Example 6 Place 50 in simple radical form. Because 50 = 25 2, we can use Property 1 to write 50 = 25 2 = 5 2. When working with square roots it can be extremely useful to recall the definition of a rational exponent. The process of placing a square root into simple radical form can be done much more efficiently using rational exponents. Taking the Square Root of an Even Power. When taking a square root of x n, when x is a positive real number and n is an even natural number, divide the exponent by two. In symbols, x n = x n/2. Also, recall that raising a product to a power requires that we raise each factor to that power. Raising a Product to a Power. (ab) n = a n b n. Let s look at an example that employs these properties. Example 7 Simplify In this example, the difficulty is the fact that the exponents are not divisible by 2. However, if possible, the first guideline of simple radical form requires that we factor out a perfect square. So, extract each factor raised to the highest possible power that is divisible by 2, as in = Now, divide each exponent by = Finally, simplify by expanding each exponential factor and multiplying = = 12 6
7 Section 5.1 Radical Notation and Rational Exponents 7 When there are variables in the radicand, then we need to be careful how we simplify. Note that if we incorrectly reduce ( 3) 2, we might be tempted to say it is 3. But recall that ( 3) 2 is a positive square root and the resulting simplification also needs to be positive. This discussion leads to the following result. The Positive Square Root of the Square of x. If x is any real number, then x 2 = x. Let s use these ideas to simplify some radical expressions that contain variables. Variable Expressions Example 8 Given that the x represents any real numbers, place the radical expression 48x 6 in simple radical form. Simple radical form demands that we factor out a perfect square, if possible. In this case, 48 = 16 3 and we factor out the highest power of x that is divisible by 2. 48x 6 = 16x 6 3 We can now use Property 1 to take the square root of each factor. 16x 6 3 = 16 x 6 3 Now, remember that the notation calls for a nonnegative square root, so we must insure that each response in the equation above is nonnegative. Thus, 16 x 6 3 = 4 x 3 3. Division Properties of Radicals We continue our review by stating the following property of radicals. Property 2 Let a and b be positive real numbers. Then, a a b =. b
8 8 Chapter 5 Simple Radical Form Continued With this property we can add two more guidelines for simple radical form. Simple Radical Form. When your answer is a radical expression: 1. If possible, factor out a perfect square. 2. Don t leave fractions under a radical. 3. Don t leave radicals in the denominator of a fraction. If x is any real number, recall again that x 2 = x. Let s look at another example. Example 9 Place the expression 18/x 6 in simple radical form. Discuss the domain. Note that x cannot equal zero, otherwise the denominator of 18/x 6 would be zero, which is not allowed. However, whether x is positive or negative, x 6 will be a positive number (raising a nonzero number to an even power always produces a positive real number), and 18/x 6 is welldefined. Keeping in mind that x is nonzero, but could either be positive or negative, we proceed by first invoking Property 2, taking the positive square root of both numerator and denominator of our radical expression x 6 = x 6 From the numerator, we factor a perfect square. In the denominator, we use absolute value bars to insure a positive square root. 18 x 6 = 9 2 x 3 = 3 2 x 3 We can use the Product Rule for Absolute Value to write x 3 = x 2 x = x 2 x. Note that we do not need to wrap x 2 in absolute value bars because x 2 is already positive. 3 2 x 3 = 3 2 x 2 x Because x could be positive or negative, we cannot remove the absolute value bars around x. We are done.
9 Section 5.1 Radical Notation and Rational Exponents 9 Radical Expressions Now that we have reviewed how to multiply and divide square roots, we will simplify a number of more extensive expressions containing square roots, Example 10 Simplify the expression (2 12)(3 3). Place your answer in simple radical form. We ll first take the product of 2 and 3, then the product of 12 and 3, then multiply these results together. (2 12)(3 3) = (2 3)( 12 3) = 6 36 Of course, 36 = 6, so we can simplify further = 6 6 = 36 Recall the following operation, where you distribute the 2, multiplying each term in the parentheses by 2. 2(3 + x) = 6 + 2x You can do precisely the same thing with radical expressions. 2(3 + 5) = Like the familiar example above, we distributed the 2, multiplying each term in the parentheses by 2. Let s look at an example. Example 11 Use the distributive property to expand the expression 12(3 + 3), placing your final answer in simple radical form. First, distribute the 12, multiplying each term in the parentheses by 12. Note that 12 3 = (3 + 3) = = However, this last expression is not in simple radical form, as we can factor out a perfect square (12 = 4 3) = 3( 4 3) + 6 = 3(2 3) + 6 =
10 10 Chapter 5 The distributive property is also responsible in helping us combine like terms. For example we know that 3x + 5x = 8x. It is the distributive property that actually provides this solution. Note how we use the distributive property to factor x from each term. 3x + 5x = (3 + 5)x Hence, 3x + 5x = 8x. You can do the same thing with radical expressions = (3 + 5) 2 Hence, = 8 2, and the structure of this result is identical to that shown in 3x + 5x = 8x. There is no difference in the way we combine these like terms. We repeat the common factor and add coefficients. In the case that we don t have like terms, as in 3x + 5y, there is nothing to be done. In like manner, each of the following expressions have no like terms that you can combine. They are as simplified as they are going to get , , and 2 x + 5 y However, there are times when it can look as if you don t have like terms, but when you place everything in simple radical form, you discover that you do have like terms that can be combined by adding coefficients. Example 12 Simplify the expression , placing the final expression in simple radical form. We can extract a perfect square (27 = 9 3) = 5( 9 3) = 5(3 3) = Note that we now have like terms that can be combined by adding coefficients = 23 3
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