Transcript Lesson1.7 Rational Exponents

Warm up
• 1. Change into Scientific Notation
• 3,670,900,000
• Use 3 significant figures
• 2. Compute: (6 x 102) x (4 x 10-5)
• 3. Compute: (2 x 10 7) / (8 x 103)
Lesson1.7 Rational Exponents
Objective:
Nth Roots
a is the square
root of b if a2 = b
a is the cube root
of b if a3 = b
Therefore in
general we can
say:
a is an nth root of
b if an = b
Nth Root
• If b > 0 then a and –a are square roots of b
• Ex: 4 = √16 and -4 = √16
• If b < 0 then there are not real number square
roots.
• Also b1/n is an nth root of b.
• 1441/2 is another way of showing √144
• ( =12)
Principal nth Root
• If n is even and b is positive, there are two
numbers that are nth roots of b.
• Ex: 361/2 = 6 and -6 so if n is even (in this case 2) and
b is positive (in this case 36) then we always choose
the positive number to be the principal root. (6).
The principal nth root of a real number b, n > 2 an integer,
symbolized by n
means an = b
b a
if n is even, a ≥ 0 and b ≥ 0
if n is odd, a, b can be any real number
index
n
radical
b
radicand
Examples
Find the principal root:
1.
5
1
2. 811/2
Evaluate
3. (-8)1/3
4. -(
1 1/4
)
16
Properties of Powers an = b and
Roots a = b1/n for Integer n>0
• Any power of a real number is a real number.
• Ex: 43 = 64
(-4)3 = -64
• The odd root of a real number is a real .
• Ex: 641/3 = 4
(-64)1/3 = -4
• A positive power or root of zero is zero.
• Ex: 0n = 0
0 1/n = 0
Properties of Powers an = b and
Roots a = b1/n for Integer n>0
• A positive number raised to an even power
equals the negative of that number raised to the
same even power.
• Ex: 32 = 9
(-3)2 = 9
• The principal root of a positive number is a
positive number.
• Ex: (25)1/2 = 5
Properties of Powers an = b and
Roots a = b1/n for Integer n>0
• The even root of a negative number is not a
real number.
• Ex: (-9)1/2 is undefined in the real number system
Warm up
• 1. 4 1
• 2. (x1/3y-2)3
• 3.
169
Practice
• (-49)1/2 =
• (-216)1/3 =
• -( 1/81)1/4 =
Rational Exponents
• bm/n = (b1/n)m = (bm)1/n
• b must be positive when n is even.
• Then all the rules of exponents apply when the
exponents are rational numbers.
• Ex: x⅓ • x ½ =
•
x ⅓+ ½ =
•
x5/6
• Ex: (y ⅓)2 = y2/3
Practice
• 274/3 =
• (a1/2b-2)-2 =
Radicals
•
is just another way of writing b1/2.
• The
is denoting the principal (positive) root
•
is another way of writing b1/n , the principal
nth root of b.
• so:
•
= b1/n = a where an = b
if n is even and b< 0, is not a real number;
if n is even and b≥ 0, is the nonnegative
number a satisfying an = b
Radicals
•
bm/n = (bm)1/n =
m
and
bm/n
=
(b1/n)m
=(
Ex: 82/3 = (82)1/3 =
= (81/3)2= (
)
m
2
)2
Practice
• Change from radical form to rational exponent
form or visa versa.
• (2x)-3/2 x>0 =
1
(2x)3/2 =
• (-3a)3/7 =
•
1
3
=
4
1
1
y4/7
= y– 4/7
3
Properties of Radicals
•
m
=(
•
m
)
2
=
•
=
•
n
= a if n is odd
•
n
= │a │if n is even
)2 = 4
=(
=
=
3
=6
=
= -2
2
= │-2│= 2
Simplify
•
=
•
=
•
•
3
6
=
=
Warm up
• 1. 36
• 2. 27
• 3. 81
3
2
4
3
3
4
Simplifying Radicals
• Radicals are considered in simplest form when:
• The denominator is free of radicals
m
•
has no common factors between m and n
m
•
has m < n
Rationalizing the Denominator
• To rationalize a denominator multiply both the
numerator and denominator by the same
radical.
• Ex:
●
=
=
2
Rationalizing the Denominator
•
•
If the denominator is a
binomial with a radical, it is
rationalized by multiplying
it by its conjugate.
(The conjugate is the same
expression as the denominator
but with the opposite sign in the
middle, separating the terms.
(√m +√n)(√m -√n)= m-n
Rationalizing the Denominator
• -9xy3
•
•
-6
+
4
-
Operations with Radicals
• Adding or subtracting radicals requires that the
radicals have the same number under the
radical sign (radicand) and the same index.
•
+
=
Operations with Radicals
• Multiplying Radicals
•
•
n
a and
5
2
xy
m
●5
b
xy
can only be multiplied if m=n.
=
5
3 2
xy
Operations with Radicals
• Simplify:
2 xy  xy
3
2
3
2 2
Sources
• http://www.onlinemathtutor.org/help/mathcartoons/mr-atwadders-math-tests/