1 Objectives To simplify radical expressions To rationalize radicals in denominators To list Pythagorean triples To apply the Pythagorean Theorem in classifying acute, right, and obtuse triangles

2 Perfect squares; radicals (square roots) Perfect Squares 4= 2 x 2 9= 3 x 3 16= 4 x 4 25= 5 x 5 36= 6 x 6 49= 7 x 7 64= 8 x 8 81= 9 x 9 100= 10 x = 11 x = 12 x = 13 x = 14 x = 15 x 15

3 Properties of radicals ≥ 0 * = x =*

4 Simplifying Radicals defined To simplify a radical means to find another expression with the same value. It does not mean to find a decimal approximation. If the number under your radical is not divisible by any of the perfect squares, your radical is in simplest form and cannot be reduced further.

5 Examples: Simplifying Radicals Find the largest perfect square under the radical OR find one perfect square under the radical at a time Find the largest perfect square under the radical

6 Rationalizing a radical in the denominator In algebra, you should not leave a radical in the denominator; to “rationalize” a radical in the denominator, we multiply by both the numerator and denominator by that radical (essentially, we multiply by 1)

7 Pythagorean Triples = = Recognizing Pythagorean triples will help save you time when solving some right triangle problems

8 Determining Acute, Right, or Obtuse for Triangles Let a, b, c be the lengths of the sides of a triangle, where c is the longest Acute: c 2 < a 2 + b 2 Right: c 2 = a 2 + b 2 Obtuse: c 2 > a 2 + b 2 a c b A triangle’s sides measure 3, 4, ? > = 25 Obtuse