close

Вход

Забыли?

вход по аккаунту

?

5.7 – Rational Exponents

код для вставкиСкачать
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,
3в€љ2 is equivalent to 21/3
4в€љ5xy is equivalent to (5xy)1/4
What about something like 5в€љx3?
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
Answer:
in radical form.
Definition of
Write
Answer:
in radical form.
Definition of
Write each expression in radical form.
a.
Answer:
b.
Answer:
Write
Answer:
using rational exponents.
Definition of
Write
Answer:
using rational exponents.
Definition of
Write each radical using rational exponents.
a.
Answer:
b.
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.
Method 2
Power of a Power
Multiply exponents.
Answer:
Evaluate
.
Method 1
Factor.
Power of a Power
Expand the square.
Find the fifth root.
Answer: The root is 4.
Method 2
Power of a Power
Multiply exponents.
Answer: The root is 4.
Evaluate each expression.
a.
Answer:
b.
Answer: 8
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.
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.
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?
Answer: 380 kg
b. If he actually lifted 379 kg, compare this to the value
predicted by the formula.
Answer: The formula prediction is slightly higher than his
actual total.
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.
Answer:
b.
Answer:
Simplify
.
Rational exponents
Power of a Power
Quotient of Powers
Answer:
Simplify.
Simplify
.
Rational exponents
Power of a Power
Multiply.
Answer:
Simplify.
Simplify
.
is the conjugate
of
Answer:
Multiply.
Simplify each expression.
a.
Answer: 1
b.
Answer:
c.
Answer:
Документ
Категория
Презентации
Просмотров
8
Размер файла
860 Кб
Теги
1/--страниц
Пожаловаться на содержимое документа