qr code generator vb.net 2010 Exponents and Roots in .NET

Creating Code128 in .NET Exponents and Roots

CHAPTER 5 Exponents and Roots
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9: 2xy2 z4 4 3x 1 z2 3 xy5 z 4 24 x4 y2 4 z4 4 33 x 1 3 z2 3 xy5 z4 16x4 y8 z16 27x 3 z6 x1 y5 z4 432x4 3 1 y8 5 z16 6 4 432x2 y13 z26 10: 2 xy4 3 yz2 4 2x3 y4 3 y4 z2 4 2x3 y12 y4 z8 2x3 y16 z8 34 x4 y4 z4 81x4 y4 z4 81x4 y4 z4 3xyz 4 2 3 4 16 4 8 4 2 2 1 12 4 2y12 z4 z x 1 y12 z4 x y y z 81 81 81 x 81x
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There are times in algebra, and especially in calculus, when you will need to 1 convert a fraction into a product Using the fact that a 1 , we can rewrite a a fraction as a product of the numerator and denominator raised to the 1 power Here is the idea: numerator numerator denominator 1 : denominator
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5x 8 5x 8 2x 3 3 2x 3 3
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1: 2: 3: 4x2 y5 2x x 3 x 1 2 x y
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CHAPTER 5 Exponents and Roots
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4: 2x 3y 2 2x 3 2x 5
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Solutions
1: 2: 3: 4: 4x2 4x2 y 5 y5 2x x 3 2x x 3 x 1 2 2 x 1 x xy 1 y 2x 2x 3y 2 2 3y 2x 3 2x 3 2x 5 1 2x 5
Roots
The square root of a number is the nonnegative number whose square is the root For example 3 is the square root of 9 because 32 9
Examples
p p 16 4 because 42 16 81 9 because 92 81 p 625 25 because 252 625 It may seem that negative numbers could be square roots It is true that p 9 is the symbol for the nonnegative number whose 3 2 9 But square is 9 Sometimes we say that 3 is the principal square root of 9 When we speak of an even root, we mean the nonnegative root In
CHAPTER 5 Exponents and Roots
p general, n a b if bn a There is no problem with odd roots being negative numbers: p 3 64 4 because 4 3 4 4 4 64: If n is even, b is assumed to be the nonnegative root Also even roots of negative numbers do not exist in the real number system In this book, it is assumed that even roots will be taken only of nonnegative numbers For p instance in x, it is assumed that x is not negative Root properties are similar to exponent properties p p p Property 1 n ab n a n b We can take the product then the root or take the individual roots then the product
Examples
p p p p 64 4 16 4 16 2 4 8 p p p p p p 5 5 5 4 6x 4 4y 4 24xy 3 4x 12x Property 1 only applies to multiplication There is no similar property for addition (nor subtraction) A common mistake is to simplify the sum of p two squares For example x2 9 x 3 is incorrect The following example should give you an idea p p expressions are not pof why these two equal If there were the property n a b n a n b, then we would have p p p p 58 49 9 49 9 7 3 10: This could only be true if 102 = 58 r p n a n a p Property 2 n b b We can take the quotient then the root or the individual roots then the quotient r p 4 2 4 p 9 9 3 Property 3 negative) p m p n a n am (Remember that if n is even, then a must not be
We can take the root then the power or the power then take the root
CHAPTER 5 Exponents and Roots
Property 4 p n p n a n an a
Property 4 can be thought of as a root-power cancellation law
Example
p p 3 3 27 33 3 p 5 2 5 p q 3 3 8x3 2x 3 2x
Practice
p 25x2 q 3 2: 8y3 1: 3: 4: q 4 x 2 q 3 5 x 1 3
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