( 3) ( 4 ) 1. Exponents and Radicals ( ) ( xy) 1. MATH 102 College Algebra. still holds when m = n, we are led to the result

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1 Exponents and Radicals ZERO & NEGATIVE EXPONENTS If we assume that the relation still holds when m = n, we are led to the result m m a m n 0 a = a = a. Consequently, = 1, a 0 n n a a a 0 = 1, a 0. Then if the relation holds even when m < n, for example, if m = 5 and n = 8, a 5 / a 8 = a 5-8 = a -3. But a 5 / a 8 = 1/a 3 and if the two results are to be consistent, a -3 = 1/a 3. In general we define Examples: = = (2x) 0 = x 0 = 2(1) = 2 5. x - y 0 = x - 1 a n 1 = n a, a 0 6. (x - y) 0 = = 1/2 8. 1/4-2 = 4 2 = x -1 = 2/x = -1-1/2 Exercises: Evaluate the following: 1. (3a) (3 + a) 0 3. (-3 2 ) ( ) ( 8 ) 7. 2 (4 ) 2 8. ( ) ( 3) ( ) ( 4 ) ( a + b )( a b ) x + y x+ y 12. ( ) ( x + y) ( x y ) 14. ( x y) ( x y ) ( ) x y 15. ( xy) 1 1

2 FRACTIONAL EXPONENTS & RADICALS In this section, the definition of a n will be extended to include all fractions or rational numbers for n. This includes the following definition of powers with fractional exponents: n m m / n The form a = a is called the principal nth root of a m. The numerator m indicates a power and the denominator n is called the root or order. a 1/n means the principal nth root of a. The symbol is called a radical, where a is the radicand and n is the index or order. Below is the analogy that exists between a power with fractional exponent and its corresponding radical: Element m / n n m a = a = n ( a) m Power Radical a Base Radicand m Numerator of exponent Exponent n Denominator of exponent Index Examples: Laws on Radicals 2

3 Simplifying Radicals There are three ways two simplify radicals: (1) removing perfect nth power; (2) reducing the index to the lowest possible order; and (3) rationalizing the denominator. A. Removing Perfect nth Power If the radicand Can be expressed as a product of two factors, one of which is a perfect nth power, where n is the index, the radical may be simplified by removing this factor and writing its nth root outside the radical as a coefficient, following the formula Examples: Simplify the following: B. Reducing the Index to the Lowest Possible Order If the radicand is an exact power of degree equal to a factor of the index, the order of the radical may be reduced by expressing the radicand as a power with fractional exponent and reducing the fractional exponent to lowest term; or you can apply the law, The radical expression is not simplified if m and n have any common factors. That is, m/n must be in simplest form. Examples: Reduce the order of the radicals: 3

4 C. Rationalizing the Denominator The process by which a fraction is rewritten so that the denominator contains only rational numbers. A variety of techniques for rationalizing the denominator are demonstrated below. A fraction that contains a radical in its denominator can be written as an equivalent fraction with a rational denominator (a denominator without a radical). Never leave a radical in the denominator of a fraction. Always rationalize the denominator. Situation 1 - Monomial Denominator When the denominator is a monomial (one term), multiply both the numerator and the denominator by whatever makes the denominator an expression that can be simplified so that it no longer contains a radical. * Sometimes the value being multiplied happens to be exactly the same as the denominator, as in this first example (Example 1): Example 1: Simplify Multiplying the top and bottom by will create the smallest perfect square under the square root in the denominator. Replacing by 7 rationalizes the denominator. * Sometimes it is necessary to multiply by whatever makes the denominator a perfect square or perfect cube or any other power that can be simplified, as seen in the next examples (Examples 2 and 3). 4

5 Example 2: Simplify Multiply by a value that will create the smallest perfect square under the radical. This will prevent the need for additional simplifications. Choosing to multiply by (and not ) will create the smallest perfect square under the radical in the denominator. Example 3: Simplify Multiplying by will create the smallest perfect cube under the radical. Replacing by 3, rationalizes the denominator. Make sure you multiply by whatever makes the radicand (the number under the radical sign) the smallest possible value to be simplified. This will avoid having to further simplify later on. 5

6 Situation 2 - More than One Term in Denominator When there is more than one term in the denominator, the process is a little tricky. You will need to multiply the numerator and denominator by the denominator's conjugate. The conjugate is the same expression as the denominator but with the opposite sign in the middle, separating the terms. Example 4: Simplify Multiply top and bottom by the conjugate of the denominator,. Notice that you are multiplying by 1, which does not change the original expression. When multiplying the denominators in this problem, distribute or use FOIL. Notice what is happening to the middle terms when you multiply the denominators. The middle terms will drop out. Also, the last term has created a perfect square under the square root. If possible, always reduce your final answer. In this problem, a factor of 2 can be removed from the top and bottom. Did you notice that in this problem, we never distributed the 2 in the numerator? It is often best to work with the denominator first, and then see what else needs to be done. 6

7 Example 5: Simplify Multiply top and bottom by the conjugate of the denominator,. In this problem, we will need to multiply out both the numerator and the denominator. Again you can use the distributive method or FOIL. Notice that while the middle terms are going to drop out on the bottom, they are not going to drop out on the top. This is OK. We just want the radical gone from the bottom. Combine terms. The answer has a radical in the numerator. This is OK. The bottom does NOT have a radical, which was our goal. Be sure to enclose expressions with multiple terms in parentheses. This will help you to remember to FOIL (or distribute) these expressions. 7

8 Situation 3 - Working with a Reciprocal When working with the reciprocal of an expression containing a radical, it may be necessary to rationalize the denominator. Example 6: Write the reciprocal of Here is our starting expression. The reciprocal is created by inverting the numerator and denominator of the starting expression. Since we now have a radical in the denominator, we must rationalize this denominator. Multiply top and bottom by the conjugate of the denominator,. Multiply the denominators in this problem, by using the distribute method or FOIL. Again, we see the middle terms dropping out. Simplify and combine. Remember, that a radical in the numerator is OK. Simplified Radical Expression A radical expression is simplified if 1. There are no radicals in a denominator. 2. There are no fractions inside a radical symbol. 3. All radicands have no nth power factors. 4. The numerator and denominator of any rational expression (fractions) have no common factors. 8

9 ADDITION AND SUBTRACTION OF RADICALS Only similar radicals can be combined in the same way as in combining like terms. Similar Radicals are radicals with the same order and the same radicand. Examples: Perform the indicated operations: Distributive property Distributive property We do not have similar radicals here, but we can Simplify Simplify the 1st expression Combine like terms First step is to rationalize the denominator of the 2 nd fraction. LCD = 15 LCD divided by each denominator then multiplied by each numerator Simplify both expressions LCD = 5 Simplify the 2 nd expression Removal of perfect nth power Combine similar terms Simplify the 1 st and 2 nd terms Combine like terms Simplify each expression Combine like terms 9

10 MULTIPLICATION AND DIVISION OF RADICALS Only radicals of the same order can be multiplied or divided. Use the following laws on radicals in finding the product or quotient of two radicals. If the radicals are of different orders, convert them first to powers with fractional exponents and change these fractional exponents into similar fractions. In case of division, if there is still a radical in the denominator, rationalizing it will be one of the steps. Examples: A. Find the product in each of the following: = 2 Simplify the radical Apply distributive property Simplify the radical Sum & difference Square of binomial Combine like terms Square of binomial Cube of binomial Sum & difference 10

11 B. Divide and simplify the following: = = 4` Exercises: Perform the indicated operations: 11