Warm up 1. Change into Scientific Notation 3,670,900,000 Use 3 significant figures 2. Compute: (6 x...
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Transcript of Warm up 1. Change into Scientific Notation 3,670,900,000 Use 3 significant figures 2. Compute: (6 x...
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 ExponentsObjective:
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 means an = b
if n is even, a ≥ 0 and b ≥ 0
if n is odd, a, b can be any real number
n bindex
radicand
radical
n b a
Examples
5 1
Find the principal root:
1.
2. 811/2
3. (-8)1/3
4. -( ) 1/41
16
Evaluate
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.
• 2. (x1/3y-2)3
• 3.
4 1
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 =
and
bm/n = (b1/n)m = ( )
Ex: 82/3 = (82)1/3 =
= (81/3)2= ( )2
m
m
2
Practice• Change from radical form to rational exponent
form or visa versa.
• (2x)-3/2 x>0 =
• (-3a)3/7 =
• 1 =
4
1 1 (2x)3/2 = 3
3
1 = y– 4/7
y4/7
Properties of Radicals• = ( ) = ( )2 = 4
• = = = 6
• = = =
• = a if n is odd = -2
• = │a │if n is even = │-2│= 2
mm
n
n
2
3
2
Simplify• =
• =
• =
• =
3
6
Warm up
2
3
36• 1.
• 2.
• 3.
3
4
27
4
3
81
Simplifying Radicals• Radicals are considered in simplest form when:
• The denominator is free of radicals• has no common factors between m and n • has m < n
m
m
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
• and can only be multiplied if m=n.
• ● =
n a m b
5 xy 5 xy2 5 xy23
Operations with Radicals
• Simplify:
332 xyxy 2 2 2
Sources• http://www.onlinemathtutor.org/help/math-
cartoons/mr-atwadders-math-tests/