5.7 – Rational Exponents Basic concept Converting from rational exponent to radical form Evaluating expressions with rational exponents Simplifying

Writing radicals in exponential form Note that (√x)2 = x = x1 Is there a way for us to represent √x using exponents? Suppose we call the power equivalent to a square root “z” Then (√x)2 = (xz)2 = x = x1 By the third law of exponents, (xz)2 = x2z Since x2z = x1, 2z = 1 and z = ½ Therefore x1/2 is equivalent to √x

More on rational exponents By similar reasoning, 3√2 is equivalent to 21/3 4√5xy is equivalent to (5xy)1/4 What about something like 5√x3? This is the 5th root of x cubed To write using rational exponents, the index goes in the denominator, and the power goes in the numerator Thus, 5√x3 = x 3/5 Similarly, √28 = 28/2 = 24 = 16 Note from the above example that rational exponents may be reduced like any other fraction

Write in radical form. Answer: Definition of Example 7-1a

Write in radical form. Answer: Definition of Example 7-1b

Write each expression in radical form. a. b. Answer: Answer: Example 7-1c

Write using rational exponents. Definition of Answer: Example 7-2a

Write using rational exponents. Definition of Answer: Example 7-2b

Write each radical using rational exponents. a. b. Answer: Answer: Example 7-2c

Evaluting expressions with radical exponents Remember, for rational exponents, the number in the numerator is an exponent while a number in a denominator is a root Usually you want to deal with the root first, then the power that was in the numerator Also recall that a negative exponent may be made positive by….? Switching the expression from numerator to denominator or vice versa See example on the next slide

Evaluate Method 1 Answer: Simplify. Example 7-3a

Method 2 Power of a Power Multiply exponents. Answer: Example 7-3b

Evaluate . Method 1 Factor. Power of a Power Expand the square. Find the fifth root. Answer: The root is 4. Example 7-3c

Method 2 Power of a Power Multiply exponents. Answer: The root is 4. Example 7-3d

Evaluate each expression. a. b. Answer: Answer: 8 Example 7-3e

Answer: The formula predicts that he can lift at most 372 kg. Weight Lifting The formula can be used to estimate the maximum total mass that a weight lifter of mass B kilograms can lift in two lifts, the snatch and the clean and jerk, combined. According to the formula, what is the maximum that U.S. Weightlifter Oscar Chaplin III can lift if he weighs 77 kilograms? Original formula Use a calculator. Answer: The formula predicts that he can lift at most 372 kg. Example 7-4a

Weight Lifting The formula Weight Lifting The formula can be used to estimate the maximum total mass that a weight lifter of mass B kilograms can lift in two lifts, the snatch and the clean and jerk, combined. Oscar Chaplin’s total in the 2000 Olympics was 355 kg. Compare this to the value predicted by the formula. Answer: The formula prediction is somewhat higher than his actual total. Example 7-4b

Weight Lifting Use the formula where M is the maximum total mass that a weight lifter of mass B kilograms can lift. a. According to the formula, what is the maximum that a weight lifter can lift if he weighs 80 kilograms? b. If he actually lifted 379 kg, compare this to the value predicted by the formula. Answer: 380 kg Answer: The formula prediction is slightly higher than his actual total. Example 7-4c

Laws of Exponents The laws of exponents hold true for expressions with rational exponents That is, the following still hold true xa * xb = xa+b xa ÷ xb = xa – b (xa)b = xab Recall that for the 1st two laws, the base of each expression must be the same Also remember that when combining fractions you often need to find a common denominator Remember – we don’t like to have negative exponents in our answers and we don’t like having radicals in denominators (sometimes necessitating rationalizing a denominator) Finally, if possible, you should reduce the index so it is a small as possible Let’s look at a few examples!

Simplify . Multiply powers. Answer: Add exponents. Example 7-5a

Simplify . Multiply by Example 7-5b

Answer: Example 7-5c

Simplify each expression. a. b. Answer: Answer: Example 7-5d

Simplify . Rational exponents Power of a Power Example 7-6a

Quotient of Powers Answer: Simplify. Example 7-6b

Simplify . Rational exponents Power of a Power Multiply. Answer: Example 7-6c

Simplify . is the conjugate of Answer: Multiply. Example 7-6d

Simplify each expression. a. b. c. Answer: 1 Answer: Answer: Example 7-6e