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6.2 Multiplying Monomials
CORD Math
Mrs. Spitz
Fall 2006
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Standard/Objectives
• Standard:
• Objectives: After studying this lesson, you should be able to:– Multiply monomials, and– Simplify expressions involving powers of
monomials.
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What’s a monomial?
• A monomial is a number, a variable, or a product of a number and one or more variables. Monomials that are real numbers are constants.
• These are monomials:-9 y 7a 3y3 ½abc5
• These are NOT monomials:
m + n x/y 3 – 4b 1/x2 7y/9z
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Notes:
• Recall that an expression of the form xn is a power. The base is x and the exponent is n. A table of powers of 2 is shown below:
20 21 22 23 24 25 26 27 28 29 210
0 2 4 8 16 32 64 128 256 512 1024
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Notes:
• Notice that each of the following is true:4 · 16 = 64 8 · 16 = 128 8 · 32 = 256
22 · 24 = 26 23 · 24 = 27 23 · 25 = 28
Look for a pattern in the products shown. If you consider only the exponents, you will find that 2 + 4 = 6, 3 + 4 = 7, and 3 + 5 = 8
These examples suggest that you can multiply powers that have the same base by adding exponents.
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Product of Powers Property
• For any number a and all integers m and n,
am · an = am+n
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Ex. 1: Find the measure of the area of a rectangle.
A = lw= x3 · x4
= x3+4
= x7x3
x4
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Ex. 2: Simplify (-5x2)(3x3y2)( xy4)
(-5x2)(3x3y2)( xy4) = (-5· 3 · )(x2 · x3 · x)(y2 · y4)
= -6x2+3+1y2+4
= -6x6y6
Step 1: Commutative and associative propertiesStep 2: Product of Powers PropertyStep 3: Simplify
2
5
2
5
2
5
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Notes: Take a look at the examples below:
(52)4 = (52)(52)(52)(52)
= 52+2+2+2
= 58
(x6)2 = (x6)(x6)
= x6+6
= x12
Since (52)4 = 58 and (x6)2 = x12, these examples suggest that you can find the power of a power by multiplying exponents.
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Power of a power
• For any number a and all integers m and n,
(am)n = amn
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Here are a few more examples
(xy)3 = (xy)(xy)(xy) = (x · x · x)(y · y · y) = x3y3
(4ab)4 = (4ab) (4ab) (4ab) (4ab) = (4 · 4 · 4 ·4)(a · a · a · a)(b · b · b · b) = 44a4b4
= 256a4b4
These examples suggest that the power of a product is the product of the powers.
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Power of a Product
• For any number a and all integers m,
(ab)m = ambm
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Ex. 3—Find the measure of the volume of the cube.
V = s3
= (x2y4)3
= (x2)3 · (y4)3
= x 2·3y4·3
= x6y12
x2y4
x2y4
x2y4
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Power of a Monomial
• For any number a and b, and any integers m, n, and p,
(ambn)p = ampbnp
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Ex. 4: Simplify (9b4y)2[(-b)2]3
(9b4y)2[(-b)2]3 = 92(b4)2y2(b2)3
= 81b8y2b6
= 81b14y2
Some calculators have a power key labeled yx . You can use it to find the powers of numbers more easily. See the next slide.
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Ex. 5: Evaluate (0.14)3
Enter: 0.14
Display will read: 0.002744, so (0.14)3 is about 0.003
yx 3 =