Chapter 5, Section 6 Radical Expressions

Radical Expressions Radical expressions are arithmetic expressions that contain a radical sign. These expressions may be simplified, added, subtracted, multiplied, or divided.

Simplifying Radical Expressions A radical expression is simplified if: the radicand contains only powers LESS THAN the index the radicand contains no fractions no radicals appear in the denominator

Example 1 Simplify First we’ll want to factor 25 into 52. Next, we’ll want to remove any number of factors equal to the index. When dividing by the index, any remainder “remains” under the radical:

Quotient Property of Radicals Recall that for n>1, and any a and b, where roots are defined and b ≠ 0,

Rationalizing the denominator Remember that there can be no radicals in the denominator, so if while we are working we GET a radical in the denominator, we get rid of it by MULTIPLYING by just enough factors of it (under a radical of the same index) to remove it, i.e. to make the exponent equal the index. We just need to make sure we multiply the numerator by the same number.

Example 2 Simplify We will apply the Quotient Property of Radicals and rationalize the denominator: Note that we don’t care if the radical is in the NUMERATOR…

Example 3 Simplify We will again apply the Quotient Property of Radicals and rationalize the denominator:

Product Property of Radicals When multiplying two radicals with the same index, just multiply the radicands: Remember that by the Symmetric Property of Equality, the reverse is true:

Example 4 Simplify We will use the Commutative and Associative properties, then apply the Product Property of Radicals. Remember to always factor any coefficients.

Adding and Subtracting Radicals Remember that we can only add and subtract like quantities, so we can only add and subtract the same radicals (same index, same radicand). E.g. We MUST simplify all radicals before we can determine if they can be added or subtracted, however.

Example 5 Simplify Even though these radicals are not the same (their radicands are different), we cannot tell if they can be combined until we simplify:

Multiplying and Dividing Radical Expressions You know how to FOIL (i.e. you know how to apply the Distributive Property!), so multiplication won’t be much of a problem. Division presents an added wrinkle because we NEVER leave radicals in the denominator. To get rid of binomials with radical expressions, we will remember that (a + b)(a - b) = a2 - b2. The numbers a + b and a - b are called conjugates of each other, and when you have a radical expression of the form a ± b in the denominator, you can get rid of it by multiplying by the conjugate of the expression.

Example 6 Simplify We see this is a binomial times a binomial, so we will FOIL it:

Example 7 Simplify This looks pretty simple already, but it has a radical in the denominator. We will multiply the denominator (and numerator!) by its conjugate to clear out the radical:

Your homework… Chapter 5 section 6, 15-47 odd, 59-81 odd, PLUS 20, 30, 34, 48, and 66. If you pursue evil with pleasure, the pleasure passes away and the evil remains; if you pursue good with labor, the labor passes away but the good remains. --Cicero