P.3 Radicals and Rational Exponents Day 3/4 8/27/18 P.3 Radicals and Rational Exponents

Principle 𝒏-th root 𝑛 𝑎 : Let 𝑛 be a positive integer greater than 1, and let 𝑎 be a real number. If 𝑎 = 0 then 𝑛 𝑎 = 0 If 𝑎 > 0 then 𝑛 𝑎 is the positive real number 𝑏 such that 𝑏 𝑛 = 𝑎 (3) (a) If 𝑎 < 0 and 𝑛 is odd, then 𝑛 𝑎 is the negative real number 𝑏 such that 𝑏 𝑛 = 𝑎 (b) If 𝑎 < 0 and 𝑛 is even, then 𝑛 𝑎 is not a real number.

𝑖𝑓 𝑎 =b, then 𝑏 2 =𝑎. 𝑖𝑓 3 𝑎 =𝑏, 𝑡ℎ𝑒𝑛 𝑏 3 =𝑎 The expression 𝑛 𝑎 is a radical, the number 𝑎 is the radicand, and 𝑛 is the index of the radical. 𝑖𝑓 𝑎 =b, then 𝑏 2 =𝑎. 𝑖𝑓 3 𝑎 =𝑏, 𝑡ℎ𝑒𝑛 𝑏 3 =𝑎

Properties of 𝑛 𝑎 : (where 𝑛 is a positive integer)

Laws of Radicals:

Warning: If 𝑎 ≠ 0 and 𝑏 ≠ 0

All possible factors have been removed from the radical Simplifying Radicals An expression involving radicals is in simplest form when the following conditions are satisfied: All possible factors have been removed from the radical 2. All fractions have radical-free denominators (accomplished by a process called rationalizing the denominator) The index of the radical is reduced. To simplify a radical, we factor the radicand into factors whose exponents are multiples of the index. The roots of these factors are written outside the radical and the “leftover” factors make up the new radicand.

Example: Evaluating Square Roots Evaluate.

                           

Rationalizing Denominators of Quotients (𝑎 > 0)

Rationalizing Denominators (continued) Radical expressions that involve the sum and difference of the same two terms are called conjugates. Thus, 𝑎 + 𝑏 and 𝑎 − 𝑏 are conjugates. If the denominator contains two terms with one or more square roots, multiply the numerator and denominator by the conjugate of the denominator.

Rationalize the denominator.

= = = =

    𝑥 1 2     2     

Radical Equations The principle of powers: If an equation 𝑎 = 𝑏 is true, then 𝑎𝑛 = 𝑏𝑛 is true for any rational number 𝑛 for which 𝑎𝑛 and 𝑏𝑛 exist. This means we will have to check our solutions in the original equation to be sure they work. If a value does not work it is called an extraneous solution and not included in the final answer