PROPERTIES OF RATIONAL

EXPONENTS

1

Section 7.2

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?!?!

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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

From Math for Artists… “These are the laws of exponents and radicals in bright, cheerful, easy to memorize colors.”

7.2 – Properties of Rational Exponents