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7.3 Square Roots and the Pythagorean Theorem Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. The Square of a Number The square of a number is the number times itself. The square of 6 is 36 because 62 = 36. The square of –6 is also 36 because (–6)2 = (–6) (–6) = 36. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 2 Square Root of a Number The reverse process of squaring is finding a square root. A square root of 36 is 6 because 62 = 36. A square root of 36 is also –6 because (–6)2 = 36. We use the symbol , called a radical sign, to indicate the positive square root of a nonnegative number. 16 4 because 42 = 16 and 4 is positive. 25 5 because 52 = 25 and 5 is positive. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 3 Square Root of a Number The square root, , of a positive number a is the positive number b whose square is a. In symbols, a b if b a. 2 9 3 because 3 9. 2 Also, 0 0. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 4 Helpful Hint Remember that the radical sign is used to indicate the positive square root of a nonnegative number. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 5 Perfect Squares 1 4 Numbers like , 36, , and 1 are 4 25 called perfect squares because their square root is a whole number or a fraction. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 6 Approximating Square Roots A square root such as 6 cannot be written as a whole number or a fraction since 6 is not a perfect square. It can be approximated by estimating, by using a table, or by using a calculator. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 7 Right Triangles One important application of square roots has to do with right triangles. A right triangle is a triangle in which one of the angles is a right angle or measures 90º (degrees). The hypotenuse of a right triangle is the side opposite the right angle. The legs of a right triangle are the other two sides. hypotenuse leg leg Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 8 Pythagorean Theorem If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then c a a b c 2 2 2 b In other words, (leg)2 + (other leg)2 = (hypotenuse)2. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Prealgebra & Introductory Algebra, 3ed 9