Chapter 5 Exponential and Logarithmic Functions 5 1

Chapter 5: Exponential and Logarithmic Functions 5. 1: Radicals and Rational Exponents Essential Question: Explain the meaning of using radical expressions

5. 1: Radicals and Rational Exponents BACKGROUND INFO (NO NEED TO COPY) �Recall that when x 2 = c (some constant), there were two solutions, and , when the constant was positive. You had no solutions when the constant was negative. ◦ x 2 = 9 → x = 3 or x = -3 �When x 3 = c, there was only one solution, , and the answer was positive or negative depending on if x was positive or negative to start. ◦ x 3 = 64 → x = 4 ◦ x 3 = -64 → x = -4 �All other higher roots act in a similar fashion

5. 1: Radicals and Rational Exponents �Solutions to xn = c ◦ When n (exponent) is even �If c > 0, one positive and one negative solution �If c = 0, one solution (x = 0) �If c < 0, no solution ◦ When n is odd �One solution �Recall help: ◦ Even # power = even # of roots ◦ Odd # power = odd # of roots

5. 1: Radicals and Rational Exponents �nth roots ◦ The nth root of c is denoted by either of the symbols: �Rules about nth roots ◦ If the outside root is the same, numbers underneath can be multiplied and divided ◦ If the number underneath the root is the same, they are like terms, and can be added or subtracted

5. 1: Radicals and Rational Exponents �Example 1: Operations on nth roots ◦ ◦ �Example 2: Evaluating nth roots ◦ Use a calculator to approximate each expression the nearest ten thousandth. ◦ ◦ , entered as “ 40^(1/5)” ≈ 2. 0913 , entered as “ 225^(1/11)” ≈ 1. 6362

5. 1: Radicals and Rational Exponents �Rational Exponents ◦ Rational exponents of the form are called nth roots. Rational exponents can also be written in the form. ◦ The numerator of the exponent represents the power a base is taken to. ◦ The denominator of the exponent represents the root. ◦ The order of application does not matter ◦

5. 1: Radicals and Rational Exponents �Assignment �Page 334 ◦ Problems 1 – 37, odd problems �Ignore the instructions about not using a calculator in problems 1 – 15 �Make sure to simplify your square roots �Show all non-calculator work �Due tomorrow

Chapter 5: Exponential and Logarithmic Functions 5. 1: Radicals and Rational Exponents (Day 2) Essential Question: Explain the meaning of using radical expressions

5. 1: Radicals and Rational Exponents �Laws of Exponents (A recap) ◦ crcs = cr+s �Multiplying same base == add exponents ◦ = cr-s �Dividing same base == subtract exponents ◦ (cr)s = crs �Power to power == multiply exponents ◦ (cd)r = crdr and �Power on outside == multiply exponents to all bases ◦ c-r = �Negative exponents move to other side of the fraction and become positive

5. 1: Radicals and Rational Exponents �Simplifying Expressions with Rational Exponents (Ex 3) ◦ Write the expression using only positive exponents ◦ Distribute the 2/3 on the outside ◦ Simplify the coefficient part & move negative exponents

5. 1: Radicals and Rational Exponents �Simplifying Expressions with Rational Exponents ◦ Example 4: ◦ Distribute… and since bases are shared, add the exponents ◦ ◦ Example 5: ◦ Take the -2 power first, then add exponents to like bases (as in Ex 4 above) ◦

5. 1: Radicals and Rational Exponents �Simplifying Expressions with Rational Exponents ◦ Ex 6: Write the expression without using radicals, using only positive exponents ◦ ◦ Get rid of the radicals. Root = denominator ◦ ◦ Multiply powers of powers. ◦ ◦ Add exponents of common bases ◦

5. 1: Radicals and Rational Exponents �Rationalizing the Numerator/Denominator ◦ Rationalizing means removing all roots from the specified side of a fraction ◦ Simple roots can be removed by multiplying top/bottom by the root. ◦ ◦ Complex roots can be removed by multiplying with the conjugate ◦ ◦ Rationalizing a numerator works the same as above

5. 1: Radicals and Rational Exponents �Assignment �Page 334 ◦ Problems 39 -77, odd problems �Make sure to simplify your square roots �Show all non-calculator work �Due whenever we get back