5.7 – rational exponents -basic concept -converting from rational exponent to radical form...

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

Write in radical form.

Answer: Definition of

Write each expression in radical form.

a.

b.

Answer:

Answer:

Write using rational exponents.

Definition of Answer:

Write using rational exponents.

Definition of Answer:

Write each radical using rational exponents.

a.

b.

Answer:

Answer:

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.

Multiply exponents.

Method 2

Answer:

Power of a Power

Answer: The root is 4.

Evaluate .

Method 1 Factor.

Power of a Power

Expand the square.

Find the fifth root.

Answer: The root is 4.

Power of a Power

Multiply exponents.

Method 2

Evaluate each expression.

a.

b.

Answer: 8

Answer:

According to the formula, what is the maximum that U.S. Weightlifter Oscar Chaplin III can lift if he weighs 77 kilograms?

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.

Original formula

Use a calculator.

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.

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.

Answer: 380 kg

Answer: The formula prediction is slightly higher than hisactual total.

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 valuepredicted by the formula.

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.

Simplify .

Multiply by

Answer:

Simplify each expression.

a.

b.

Answer:

Answer:

Simplify .

Rational exponents

Power of a Power

Quotient of Powers

Answer: Simplify.

Simplify .

Rational exponents

Power of a Power

Answer: Simplify.

Multiply.

Answer: Multiply.

Simplify .

is the conjugate

of

Answer: 1

Simplify each expression.

a.

b.

c. Answer:

Answer: