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Fundamentals of Mathematics 1.4 Exponents and Radicals (Continue) Ricky Ng Lecture 7 September 11, 2013 Ricky Ng Fundamentals of Mathematics Announcements Registration for Test 1. Fingerprint in CASA. Online Quiz 1 due 9/11 Wednesday. Online Quiz 2 & 3 due 9/13 Friday. First homework is due on 9/12. Second homework is due on 9/14. Ricky Ng Fundamentals of Mathematics Last Time... xm ⇥ xn = xm+n (xm )n = xmn x0 = 1 xn ⇥ y n = (xy)n x n = 1 xn Ricky Ng Fundamentals of Mathematics Recap Recall that xn xm can be view as xn ⇥ 1 = xn ⇥ x xm m xm = xm xn n = xn m . So, Rule Ricky Ng Fundamentals of Mathematics Fractions with Exponents There are two rules to know when dealing with fractions with exponents Rule Given non-zero real numbers x and y, and integer n ✓ ◆n x xn = n; y y and ✓ ◆ x y Ricky Ng 1 = y . x 1.4 Exponents and Radicals Why?? Ricky Ng 1.4 Exponents and Radicals Examples Evaluate the following: 3 5 2 1 3 2 ⇥ 3 4 1 Ricky Ng 1.4 Exponents and Radicals Popper 2: Question 1 Simplify the following: ✓ ◆ 3 2 2 a) 94 b) 49 c) 34 d) 29 e) None of the above Ricky Ng 1.4 Exponents and Radicals Caution When x > 0, then n ( x) = ( xn , xn , n is even; n is odd. Why? Think about ( 1)n ... Ricky Ng 1.4 Exponents and Radicals More Examples Simplify the following with no negative exponents: (2x3 y 2 z 5 )2 a0 +b0 (a+b)0 Ricky Ng 1.4 Exponents and Radicals 2x6 y 6x2 y 3 ⇣ a2 b 5 a 1 b2 2 ⌘ 3 Ricky Ng 1.4 Exponents and Radicals Popper 2: Question 2 Simplify the following with no negative exponent: (x2 y 1 )2 ⇥ x 4 y a) y 3 b) y1 c) 1 d) x3 e) y2 x4 Ricky Ng 1.4 Exponents and Radicals Square Roots Definition (Square root) A real number y is called a square root of x if y 2 = x. In general if x > 0, then x has two sqaure roots, one is negative and one is positive. The principal square root of x is the positive square root, p and we denote it by x. Remark p If x > 0, we write x for its negative square root, and p ± x to denote both positive and negative square roots. p Note that 0 = 0, so 0 only has one square root. p If x < 0, then x is not a real number. In this class, we only deal real numbers, so we do not allow that. Ricky Ng 1.4 Exponents and Radicals We also write x1/2 = This is because p x. p (x1/2 )2 = ( x)2 = x, which agrees with the exponent rules. Remark In other words, square root can be seen as a fraction p exponent. In fact, this is more useful than · in calculations, as we shall see later today. Ricky Ng 1.4 Exponents and Radicals Examples Find p 64. Find square roots of 81. Ricky Ng 1.4 Exponents and Radicals Since p · behaves like exponents, the exponent rules apply: Rule For x, y > 0, p p p x y = x1/2 y 1/2 = (xy)1/2 = xy, and s✓ ◆ ✓ ◆ p x x 1/2 x1/2 x = = 1/2 = p . y y y y Ricky Ng 1.4 Exponents and Radicals Examples Find p Find q 3⇥ p 12. 81 16 . Ricky Ng 1.4 Exponents and Radicals Popper 2: Question 3 Evaluate the square roots of 81. (Note that I did not write p 81...) a) b) c) d) 9 -9 ±9 Let me grab my calculator... Ricky Ng 1.4 Exponents and Radicals Popper 2: Question 4 Evaluate a) b) c) d) e) p 2 ⇥ (8)1/2 4 ±4 8 ±8 8 Ricky Ng 1.4 Exponents and Radicals Radicals Here are some terminology... Definition (Radical) For x > 0 and n a natural number, we write p x1/n = n x to denote its nth -root of x. We call x a radicand, the symbol p · the radical, and n the degree. Definition (Perfect square) p If both x and x are natural numbers, then we call x a perfect square. For example, 4 = 22 , 49 = 72 are perfect square; but 51 is not. Ricky Ng 1.4 Exponents and Radicals Simplest Radical Form Sometimes p x can be further simplified. More precisely, Definition p We say x is in the simplest radical form if no factor of x is a perfect square. Example p 6 is in simplest radical form. p But 32 is not. Why? Because 32 = 2 ⇥ 16, and 16 = 42 is a perfect square. Ricky Ng 1.4 Exponents and Radicals Question p If x is not in the simplest radical form, how do we simplify further? Rule In this case, we can write x= a2 |{z} perfect square ⇥ b |{z} . some other # Then, using the product rule p p p p p x = a2 ⇥ b = a2 · b = a b p If b is again not in the simplest radical form, repeat. Ricky Ng 1.4 Exponents and Radicals Examples Write each of the following in the simplest radical form, or as a rational number. p 24 541/2 Ricky Ng 1.4 Exponents and Radicals q 80 9 Ricky Ng 1.4 Exponents and Radicals Popper 2: Question 5 Write p 48 in the simplest radical form. a) b) c) d) e) p 4 3 p 3 8 6 p 3 6 p 4 2 Ricky Ng 1.4 Exponents and Radicals