Copyright © Cengage Learning. All rights reserved. Roots, Radical Expressions, and Radical Equations 8

Copyright © Cengage Learning. All rights reserved. Section 8.1 Square Roots and the Pythagorean Theorem

3 Objectives Find the principal square root of a perfect square. Use a calculator to find an approximation of a square root to a specified number of decimal places. Determine whether a number is rational, irrational, or imaginary. Graph a square root function

4 Objectives Apply the Pythagorean theorem to find the length of an unknown side of a right triangle. 5 5

5 Square Roots and the Pythagorean Theorem In this section, we will reverse the squaring process and find square roots of numbers. We also will discuss an important theorem about right triangles, called the Pythagorean theorem.

6 Find the principal square root of a perfect square 1.

7 Principle Square Root of a Perfect Square To find the area A of the square shown in Figure 8-1, we multiply its length by its width. A = 5  5 = 5 2 = 25 The area is 25 square inches. Figure 8-1

8 Principle Square Root of a Perfect Square We have seen that the product 5  5 can be denoted by the exponential expression 5 2, where 5 is raised to the second power. Whenever we raise a number to the second power, we are squaring it, or finding its square. This example illustrates that the formula for the area of a square with a side of length s is A = s 2.

9 Square Root of a Perfect Square Here are additional perfect squares of numbers. The square of 3 is 9, because 3 2 = 9. The square of –3 is 9, because (–3) 2 = 9. The square of is, because. The square of 0 is 0, because 0 2 = 0.

10 Principle Square Root of a Perfect Square Suppose we know that the area of the square shown in Figure 8-2 is 36 square inches. To find the length of each side, we substitute 36 for A in the formula A = s 2 and solve for s. A = s 2 36 = s 2 To solve for s, we must find a positive number whose square is 36. Figure 8-2

11 Principle Square Root of a Perfect Square Since 6 is such a number, the sides of the square are 6 inches long. The number 6 is called a square root of 36, because 6 2 = 36. Here are additional examples of square roots. 3 is a square root of 9, because 3 2 = 9. –3 is a square root of 9, because (–3) 2 = 9. is a square root of, because. 0 is a square root of 0, because 0 2 = 0.

12 Principle Square Root of a Perfect Square In general, the following is true. Square Roots The number b is a square root of a if b 2 = a. All positive numbers have two square roots—one positive and one negative. We have seen that the two square roots of 9 are 3 and –3. The two square roots of 144 are 12 and –12, because 12 2 = 144 and (–12) 2 = 144. The number 0 is the only number that has just one square root, which is 0.

13 Principle Square Root of a Perfect Square The symbol, called a radical sign, is used to represent the positive (or principal) square root of a number. Any expression involving a radical sign is called a radical expression. Principal Square Roots If a > 0, the radical expression represents the principal (or positive) square root of a. The principal square root of 0 is 0: = 0. The expression under a radical sign is called a radicand. For example, in the radical expression, x is the radicand.

14 Principle Square Root of a Perfect Square The principal square root of a positive number is always positive. Although 3 and –3 are both square roots of 9, only 3 is the principal square root. The symbol represents 3. To represent –3, we place a – sign in front of the radical: = 3 and = –3 Likewise, = 12 and = –12

15 Example Find each square root. a. b. c. d. e. f.

16 Example 1 g. h. cont’d

17 Use a calculator to find an approximation of a square root to a specified number of decimal places 2.

18 Use a calculator to find an approximation of a square root to a specified number of decimal places Square roots of certain numbers such as 7 cannot be computed. However, we can find an approximation of with a calculator.

19 Example The period of a pendulum is the time required for the pendulum to swing back and forth to complete one cycle. (See Figure 8-3.) The period t (in seconds) is a function of the pendulum’s length l (in feet), which is defined by t = f (l ) = 1.11 Find the period of a pendulum that is 5 feet long. Figure 8-3

20 Example – Solution We substitute 5 for l in the formula and simplify. t = 1.11 = 1.11  The period is approximately 2.5 seconds for a 5-foot-long pendulum.

21 Determine whether a number is rational, irrational, or imaginary 3.

22 Rational, Irrational, or Imaginary Numbers such as 4, 9, 16, and 49 are called integer squares, because each one is the square of an integer. The square root of any integer square is an integer, and therefore a rational number: and Square roots of positive integers that are not integer squares are irrational numbers. For example, is an irrational number.

23 Determine whether a number is rational, irrational, or imaginary Recall that an irrational number is a nonrepeating, nonterminating decimal. Comment Square roots of negative numbers are not real numbers. For example, is nonreal, because the square of no real number is –4. The number is an example from a set of numbers called imaginary numbers.

24 Rational, Irrational, or Imaginary We will assume that all variable radicands under square root symbols are either positive or 0. Thus, all square roots will be real numbers.

25 Example Determine whether the following are rational, irrational, or imaginary. a. b. c. Solution: a. is rational because

26 Example – Solution b. is irrational because 11 is not an integer square. c. is imaginary because there is no real number whose square is –9. cont’d

27 Graph a square root function 4.

28 Graphing a Square Root Function Since there is one principal square root for every nonnegative real number x, the equation f (x) = determines a square root function.

29 Example Graph: Solution: To graph this function, we make a table of values and plot each pair of points. The graph appears in Figure 8-4(a). Figure 8-4(a)

30 Example 4 – Solution A calculator graph appears in Figure 8-4(b). Figure 8-4(b) cont’d

31 Apply the Pythagorean theorem to find the length of an unknown side of a right triangle 5.

32 Applying the Pythagorean Theorem A triangle that contains a 90  angle is called a right triangle. The longest side of a right triangle is the hypotenuse, which is the side opposite the right angle. The remaining two sides are the legs of the triangle. In the right triangle shown in Figure 8-5, side c is the hypotenuse, and sides a and b are the legs. Figure 8-5

33 Graphing a Square Root Function The Pythagorean theorem provides a formula relating the lengths of the sides of a right triangle. Natural numbers that satisfy this theorem are called Pythagorean triples. The Pythagorean Theorem If the length of the hypotenuse of a right triangle is c and the lengths of the two legs are a and b, then c 2 = a 2 + b 2

34 Graphing a Square Root Function Square Root Property of Equality Suppose a and b are positive numbers. If a = b, then. Since the lengths of the sides of a triangle are positive numbers, we can use the square root property of equality (and only consider the positive root) and the Pythagorean theorem to find the length of an unknown side of a right triangle when we know the lengths of the other two sides.

35 Example – Building A High-Ropes Adventure Course The builder of a high-ropes course wants to stabilize the pole shown in Figure 8-6 by attaching a cable from a ground anchor 20 feet from its base to a point 15 feet up the pole. How long will the cable be? Figure 8-6

36 Example – Solution We can use the Pythagorean theorem, with a = 20 and b = 15. c 2 = a 2 + b 2 c 2 = c 2 = c 2 = 625 = c = 25 The cable will be 25 feet long. Substitute 20 for a and 15 for b. Square each term = 625 Take the positive square root of both sides. and, because c  c = c 2