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Section 10.1: Radical Expressions And Graphs §1 Find Square Roots By definition, a number b is called a square root of a if b2 a . We use the radical symbol √ to denote the square root of a number. Hence, if 62 36, then the square root of 36 is 6. Note also that (6)(6) 36 , so 6 is a square root of 36. Hence a positive number always has two square roots – the positive or principal square root, and the negative square root. It’s very important to note, however, that the radical symbol √ always represents the positive square root. Hence we never say that 36 6 ; we say that 36 6 only. The number inside the radical symbol is called the radicand and the entire expression is called the radical. If we square a radical, the result is the radicand itself. Hence something like with a negative expression. Something like 4 2 4 2 4 . You must be careful 4 , not -4. PRACTICE 1) Find 25 and 144 and 2) Find the square of 9 16 3 and the square of 6 §2 Irrational Numbers And Perfect Squares Any number whose square root is rational is called a perfect square. It’s a very good idea to memorize the first 20 perfect squares. For example, since 152 225, then 225 15 . So 225 is called a perfect square. A number that is not rational is called irrational. Many square roots are irrational, like number is neither rational or irrational, then it is not a real number, like 3 and 10 . If a 4 . §3 Finding Higher Roots Remember, if a b, then b2 a . Note that the power of b is 2. That’s why the radical symbol is called the square root. We see that finding the square root of a number is the opposite (or inverse) of squaring a number. But what about if we cube a number? Take something like 43 64 . What would be its inverse? Note that here the power is 3. Hence the inverse is not a square root, we call it the cube root. Hence if 43 64, then 3 64 4 . More generally, if a n b, then n b a . In the radical form, the number n is called the index or order of the radical. This is a term you definitely want to know. PRACTICE 3 8 3) Find 3 8 and 4) Find 4 16 and 4 16 §4 Finding nth Roots Of nth Powers The expression a 2 gets to be a little tricky. For example, what is the difference between Nothing, actually. They both equal 6. However, a lot of people think that reality, we say that 62 and (6) 2 ? (6)2 6 . This is not the case. In a 2 a . This is telling us that in this case, a may be a negative number. For higher indexes, the same property applies but with a slight difference if the index is even or odd. If n is a positive even integer, then n a n a , while if n is a positive odd integer, n an a . PRACTICE 5) Simplify 4 (5)4 and 6) Simplify 4 28 3 (5)3 NOTE: We will not go over the section on graphing or using a calculator to find roots.