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# 5.7 – Rational Exponents

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5.7 вЂ“ Rational Exponents
-Basic concept
-Converting from rational exponent to
-Evaluating expressions with rational
exponents
-Simplifying
вЂў Note that (в€љx)2 = x = x1
вЂў Is there a way for us to represent в€љx using
exponents?
вЂў Suppose we call the power equivalent to a
square root вЂњzвЂќ
вЂў Then (в€љx)2 = (xz)2 = x = x1
вЂў By the third law of exponents, (xz)2 = x2z
вЂў Since x2z = x1, 2z = 1 and z = ВЅ
вЂў Therefore x1/2 is equivalent to в€љx
More on rational exponents
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By similar reasoning,
3в€љ2 is equivalent to 21/3
4в€љ5xy is equivalent to (5xy)1/4
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
вЂў Thus, 5в€љx3 = x 3/5
вЂў Similarly, в€љ28 = 28/2 = 24 = 16
вЂў Note from the above example that rational exponents
may be reduced like any other fraction
Write
Definition of
Write
Definition of
Write each expression in radical form.
a.
b.
Write
using rational exponents.
Definition of
Write
using rational exponents.
Definition of
Write each radical using rational exponents.
a.
b.
exponents
вЂў Remember, for rational exponents, the number
in the numerator is an exponent while a number
in a denominator is a root
вЂў Usually you want to deal with the root first, then
the power that was in the numerator
вЂў Also recall that a negative exponent may be
вЂў Switching the expression from numerator to
denominator or vice versa
вЂў See example on the next slide
Evaluate
Method 1
Simplify.
Method 2
Power of a Power
Multiply exponents.
Evaluate
.
Method 1
Factor.
Power of a Power
Expand the square.
Find the fifth root.
Method 2
Power of a Power
Multiply exponents.
Evaluate each expression.
a.
b.
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 ChaplinвЂ™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?
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
вЂў The laws of exponents hold true for expressions with
rational exponents
вЂў That is, the following still hold true
вЂў xa * xb = xa+b
вЂў xa Г· xb = xa вЂ“ b
вЂў (xa)b = xab
вЂў Recall that for the 1st two laws, the base of each
expression must be the same
вЂў Also remember that when combining fractions you often
need to find a common denominator
вЂў Remember вЂ“ we donвЂ™t like to have negative exponents in
denominators (sometimes necessitating rationalizing a
denominator)
вЂў Finally, if possible, you should reduce the index so it is a
small as possible
вЂў LetвЂ™s look at a few examples!
Simplify
.
Multiply powers.
Simplify
.
Multiply by
Simplify each expression.
a.
b.
Simplify
.
Rational exponents
Power of a Power
Quotient of Powers
Simplify.
Simplify
.
Rational exponents
Power of a Power
Multiply.
Simplify.
Simplify
.
is the conjugate
of
Multiply.
Simplify each expression.
a.
b.