Square Roots: The square root of a number is one of its two equal factors.

1. Square Roots: The square root of a number is one of its two equal factors. (Using symbols: a is the square root of b if a2 = b.) Example: The square root of 36 is 6 since 66 = 36. The square root of 36 is also – 6, since (– 6)  (– 6) = 36. Note: Negative real numbers do not have square roots because any nonzero real number is positive when squared. (No number multiplied by itself will give a negative real number.) The number under the radical symbol is called the radicand. (49 and 81 are the radicands.) Example 1. Simplify the following radical expressions. Answers: Your Turn Problem #1 Simplify the following radical expressions. The positive square root is called the principle square root. We will mainly be concerned with the principle square root.

2. These examples are read: “the cube root of 8 is 2” “the cube root of –125 is –5” In general, x is a cube root of of y if x3=y. Also note, the cube root of negative number is a negative number. Example 2. Simplify the following radical expressions. Answers: Your Turn Problem #2 Simplify the following radical expressions. Answers:

3. Thus far, we have covered square roots and cube roots. There are also nth roots: 4th roots, 5th roots, etc. Examples: The numbers which designate the root is called the index #. The index # for the square root is a 2. However it is not usually written. Example 3. Simplify the following radical expressions. Answers: Your Turn Problem #3 Simplify the following radical expressions. Answers: Note: If the index # is even, there is no real number for the nth root of a negative number.

4. Properties of Radicals Recall that negative numbers don’t have real number square roots. It is also true that negative numbers don’t have real number nth roots if n is an even number. For the following properties, we’ll assume that the radicand is positive for any even number index. 1st, some examples where the radical expression is raised to a power equal to the index #. A few more examples where the radicand has an exponent equal to the index. Writing Radical Expressions in Simplest Radical Form Let’s make some observations, then we can state another property.

5. The factors of 40 are: The largest perfect square is 4. So we will rewrite the square root using the 4 and 10. The factors of 72 are: Now replace the square root of 36 with 6 and we’re done. The largest perfect square is 36. So we will rewrite the square root using the 36 and 2. Procedure: Writing Radical Expressions in Simplest Radical Form: Write the square root as a product of two square roots where one of the radicands is the largest perfect square that divides evenly into the original number. Then replace the square root with the whole number it is equal to. Leave as multiplication. Note: examples of perfect squares are 1, 4, 9, 16, 25, 36, 49, etc. Now replace the square root of 4 with 2 and we’re done.

6. Another Method for Writing Square Roots in Simplest Radical Form Some students have a difficult time with the previous method. This method is a little more writing but the process is more straightforward. 1. Find the prime factorization of the given radicand. 2. Circle the pairs. 3. For every pair, one of the circled numbers will be written in front of the radical. Whatever numbers are not circled stay under the radical. (Multiply if more than one number.) Your Turn Problem #4 Simplify the following radical expressions. Answers: 1. Write the prime factorization of 40. 2. Circle the pairs (only a pair of 2’s). 3. One 2 is written in front, the 2 and 5 remain inside. 1. Write the prime factorization of 72. 2. Circle the pairs (pair of 2’s & 3’s). 3. The 2 & 3 are written in front, the 2 remains inside.

7. Writing Cube Roots in Simplest Radical Form Method 1. Write the cube root as a product of two cube roots where one of the radicands is the largest perfect cube that divides evenly into the original number. Then replace the cube root with the integer it is equal to. Leave as multiplication. Note: examples of perfect cubes are 1, 8, 27, 64, 125, etc. Method 2. Again, many students have a difficult time with method 1. This method is a little more writing but the process is more straightforward. 1. Find the prime factorization of the given radicand. 2. Circle the groups of 3 equal factors. 3. For every group of 3, one of the circled numbers will be written in front of the radical. Whatever numbers are not circled stay under the radical. (Multiply if more than one number.) Rewrite the 270 with 27 and 10 since 27 is a perfect cube. Then replace the cube root of 27 with 3. Rewrite the 56 with 8 and 7 since 8 is a perfect cube. Then replace the cube root of 8 with 2. 1. Write the prime factorization of 56. 1. Write the prime factorization of 72. 2. Circle the groups of 3 equal factors. 2. Circle the groups of 3 equal factors. 3. One 2 is written in front, the 7 stays inside. 3. The 3 is written in front, the 2 & 5 stay inside.

8. Your Turn Problem #5 Simplify the following radical expressions. (Write in simplest radical form.) Answers: Simplifying Square Roots that Involve Fractions We will now need the following property: Property for Simplifying Radical Expressions that Involve Quotients. In general,

9. Answers: Separate into the square root of the numerator divided by the square root of the denominator. Then simplify each (write both in simplest radical form). Separate into the square root of the numerator divided by the square root of the denominator. Then simplify each (write both in simplest radical form). Your Turn Problem #6 Simplify the following radical expressions.

10. Rationalizing the Denominator (Square Roots) In the last example, the denominators were perfect square roots. The numerator still contained a radical but not the denominator. A rational expression (a fraction) is not considered simplified if it contains a radical in the denominator. The process of “rationalizing the denominator” will take care of this. Observe the following: Procedure: Rationalizing the denominator of a square root. (If the denominator contains a non-perfect square root) 2. Then simplify. Next Slide If a square root is multiplied by itself, the result is the radicand (without square root).

11. Your Turn Problem #7 Simplify the following radical expressions. Answers: Separate into the square root of the numerator divided by the square root of the denominator. Then multiply the denominator by itself and multiply the numerator by the same number. This can be simplified differently. The denominator can be written in simplest radical form 1st before multiplying by itself.

12. Rationalizing the Denominator (Cube Roots) A perfect cube has 3 equal factors. Observe the following: If a cube root is multiplied by itself, the result is not a whole number. Procedure: Rationalizing the denominator of a cube root. (If the denominator contains a non-perfect cube root) 2. The denominator should be a whole number. Write the numerator in simplest radical form. Reduce the fraction if possible. Next Slide 1. Multiply the denominator by another cube root which will make it a perfect cube root (i.e. 8, 27, 125, etc). Whatever we multiply by the denominator, we need to multiply by the numerator.

13. Answer: Your Turn Problem #8 Simplify the following radical expressions. Answers: The End B.R. 10-16-06