Chapter 5: Exponential and Logarithmic Functions 5.1: Radicals and Rational Exponents

1. Chapter 5: Exponential and Logarithmic Functions5.1: Radicals and Rational Exponents Essential Question: Explain the meaning of using radical expressions

2. 5.1: Radicals and Rational Exponents BACKGROUND INFO (NO NEED TO COPY) • Recall that when x2 = c (some constant), there were two solutions, and , when the constant was positive. You had no solutions when the constant was negative. • x2 = 9 → x = 3 or x = -3 • When x3 = c, there was only one solution, , and the answer was positive or negative depending on if x was positive or negative to start. • x3 = 64 → x = 4 • x3 = -64 → x = -4 • All other higher roots act in a similar fashion

3. 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

4. 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. 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

6. 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

7. 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

8. Chapter 5: Exponential and Logarithmic Functions5.1: Radicals and Rational Exponents (Day 2) Essential Question: Explain the meaning of using radical expressions

9. 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

10. 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

11. 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)

12. 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

13. 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

14. 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