properties of rational exponents 1 section 7.2. 7.2 – properties of rational exponents simplifying...
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7.2 – Properties of Rational Exponents
Simplifying Expressions Containing Rational Exponents:
Laws of Exponents: For any integers m, n (assuming no divisions by 0)
m n m nx x x
nm mnx x
n n nxy x yn n
n
x x
y y
0 1x
mm n
n
xx
x
1n na a m
nm
n m na a a new!
new!
1nn
xx
n n
x y
y x
1 nn
xx and
Recognize these?!?!
3
m n m nx x x
n n nxy x y
mm n
n
xx
x
n n
n
x x
y y
nm mnx x
1nn
xx
n nx y
y x
7.2 – Properties of Rational Exponents
Properties of Exponents Applied to Radicals:
Simplifying Radicals: A radical is in simplest form when…
No radicals appear in the denominator of a fraction The radicand cannot have any factors that are perfect roots
(given the index)
Examples: Simplify each expression.
n n nab a bn
nn
a a
b b m
n m na a
12 50 3 16
7.2 – Properties of Rational Exponents
Simplifying Radical Expressions Containing Variables:
Examples: Simplify each expression. Assume that all variables are positive.
5 5x
84 16x
7b
6 53 54x y
When we divide the exponent by the index, the remainder remains
under the radical
7.2 – Properties of Rational Exponents
Multiplying and Dividing Radical Expressions:
Examples: Simplify each expression. Assume that all variables are positive.
35 20x x
23
4 23
3
81
xy
x y
43 3 10
we will use: n n nab a b
we will use: n
naan
b b
we will use: m
n m na a
7.2 – Properties of Rational Exponents
Rationalizing Denominators: Recall that simplifying a radical expression means that no radicals appear in the denominator of a fraction.
Examples: Simplify each expression. Assume that all variables are positive.
24
5
5
4 2
3
4
2
7.2 – Properties of Rational Exponents
Adding and Subtracting Radical Expressions: simplify each radical expression combine all like-radicals
(combine the coefficients and keep the common radical)
Examples: Simplify each expression. Assume that all variables are positive.
125 20
2 12 3 27
2 2 3 338 25 8xy x y x y
544 32 2x x
7.2 – Properties of Rational Exponents
Examples: Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive.
2 1 13 2 4x x x
344 8x y
1124
34
2 2
2
xy x y
x y
7.2 – Properties of Rational Exponents
Examples: Simplify each expression. Assume that all variables are positive.
5 8 3 3
2 2x x
2
2 3 5
4 2 3 5 2 8
7.2 – Properties of Rational Exponents
Example: The final velocity, v, of an object in feet per second (ft/sec)
after it slides down a frictionless inclined plane of height h feet is:
where is the initial velocity
in ft/sec of the object.
What is the final velocity, v, of an object that slides down a frictionless inclined plane of height 2 feet with an initial velocity of 4 ft/sec?
2064v h v 0v
7.2 – Properties of Rational Exponents
Homework: pgs. 411-412 #22-28 even, 34-36, 42, 44, 50, 52, 56, 58
7.2 – Properties of Rational Exponents