1. 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. 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 10 121= 11 x 11 144= 12 x 12 169= 13 x 13 196= 14 x 14 225= 15 x 15

3. 3 Properties of radicals ≥ 0 * = x =*

4. 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. 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. 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. 7 Pythagorean Triples 3 2 + 4 2 = 5 2 5 2 + 12 2 = 13 2 3 4 56 8 10 9 12 15 5 12 1310 24 2615 36 39 7 24 2514 48 50 8 15 1716 30 34 9 40 4118 80 82 Recognizing Pythagorean triples will help save you time when solving some right triangle problems

8. 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, 6 6 2 ? 3 2 + 4 2 36 > 9 + 16 = 25 Obtuse