These statements are illustrated by the square at the

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These statements are illustrated by the square at the right, with side length 5 and area 5 2 = 25 . As stated in the introduction, we will usually not mention specific units of measurement. Now let’s go in reverse. Suppose we know the area A and would like to figure out the side length s . In other words, we need to solve the equation s 2 = A for the side length s , which must be a positive real number. Then s = A, the square root of A. For example, the side length of a square with area 9 is 9 = 3 . In general, the side length will be irrational. For example, the side length of a square with area 2 is 2 1 . 414 while the side length of a square with area 3 is 3 1 . 732 . A similar discussion applies to a cube with side length s . Its volume is V = s 3 . If we know V and want to find s , then s = 3 V is called the cube root of V. Stanley Ocken M19500 Precalculus Chapter 1.2: Exponents and Radicals
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Welcome Whole number powers Rewriting Fractions Radicals and fractional exponents Simplifying radicals. Exercises Quiz Review Radical notation and fractional exponents In the last slide, the powers and roots involved were positive real numbers, used for measurement. But it’s tricky to extend the discussion to negative numbers. The equation x 2 = 4 has two solutions x = 2 and x = - 2 , written x = ± 2 . The square root of 4 is the positive solution. Thus 4 = 2 , not ± 2 . The equation x 2 = - 4 has no solutions, since the square of any real number is positive. Thus - 4 is undefined. Squares and square roots For all real numbers u > 0 , the equation x 2 = u has two solutions: x = u (positive) and x = - u (negative). For all real numbers u < 0 , x 2 = u has no solutions and so u is undefined. The situation is completely different when we deal with third powers and roots. Cubes and cube roots For all real numbers u , the equation x 3 = u has one and only one solution x = 3 u , which has the same sign as u . Stanley Ocken M19500 Precalculus Chapter 1.2: Exponents and Radicals
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Welcome Whole number powers Rewriting Fractions Radicals and fractional exponents Simplifying radicals. Exercises Quiz Review Example 10: 4 = 2 The solution of x 2 = 4 is x = ± 4 = ± 2 . However, x 2 = - 4 has no solutions. x 3 = 8 has one solution x = 3 8 = 2 . x 3 = - 8 has one solution x = 3 - 8 = - 2 . Similar statements hold for solutions of x n = u depending on whether n is even or odd. For n > 3 , we call n u the n th root of u . Even powers and roots Suppose n is even. For all real numbers u > 0 , the equation x n = u has two solutions: x = n u (positive) and x = - n u (negative). For all real numbers u < 0 , x n = u has no solutions and so n u is undefined. Odd powers and roots Assume n is odd. For all real numbers u , the equation x n = u has one and only one solution x = n u , which has the same sign as u .
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