Intermediate Algebra chapter 8

Roots and Radical
Chapter 8, 

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Roots and Radicals

A radical expression is .

  is the radical sign, a is the radicand, and m is the index.

  means the square root of a.

The principal (or positive) square root of the number a is written   and is equal to the positive number whose square equals a.

For this class, "square root" means principal square root.

  means the cube root of a.

  means the m^th root of a.

    if  b^m = a.

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Even Indexes

The m^th root of a,   ,  where m is an even index and a is a positive real number, is thatpositive real number b such that  b^m = a.

Example:    =  3 since 3⁴ = 81.

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Odd Indexes

The m^th root of a,   ,  where m is an odd index and a is any real number, is that real number b such that  b^m = a.

Example:    =  4 since 4³ = 64.

Example:    =  -4 since (-4)³ = -64.

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Square Roots of Something Squared

Consider any real number squared. It's square root is the absolute value of that real number.

Example:    =  |a|

Example:    =  |-6|  =  6

Example:    =  |x - 2|

Rational Exponents

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For the rest of this chapter, it will be assumed that all variables represent non-negative real numbers. So,

  =  a, a    0.

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Radical to Exponential Form

For a    0 and n > 0
  =  a^1/n  =  a^m/n
  =  a^n/n  =  a

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Exponential to Radical Form

For a    0 and n > 0
a^m/n  =  

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Rules of Exponents

The rules of exponents are the same
The rules of exponents apply to rational numbers in the exponents.

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Factoring

When factoring out the greatest common factor, take out the smallest exponential power of any factors that are common to all terms.

Example:

x-2 + x2 = x-2(1 + x4) =

1 + x4 x2

Also, look for expressions that are in quadratic form, that are then easily factored. 

Example:
	x4/3 + 2x2/3 + 1
=	(x2/3)2 + 2(x2/3) + 1
	Letting y = x2/3, then
=	y2 + 2y + 1
=	(y + 1)2
	Substituting x2/3 back in for y, gives
=	(x2/3 + 1)2

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Complex Numbers
Chapter 8,

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Imaginary Numbers

The imaginary unit is and is denoted by i, and

By definition,
i²   =   -1.

For any positive real number n,

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Complex Numbers

Complex numbers have the form
a + bi where a and b are real numbers.

Examples:

• 7 + 3i
• 2 - 5i
• -1 - 4i

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Adding and Subtracting Complex Numbers
(a + bi) + (c + di) = (a + c) + (b + d)i

Steps to adding and subtracting complex numbers:

1. Change all imaginary numbers to bi form.
2. Add (or subtract) the real parts of the complex numbers.
3. Add (or subtract) the imaginary parts of the complex numbers.
4. Write the answer in the form a + bi.

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Multiplying Complex Numbers
(a + bi)(c + di) = (ac - bd) + (ad + bc)i

Steps to multiplying complex numbers:

1. Change all imaginary numbers to bi form.
2. Multiply the complex numbers as you would multiply polynomials.
3. Substitute -1 for each i².
4. Combine the real parts and the imaginary parts.
5. Write the answer in the form a + bi.

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Complex Conjugates

The complex conjugate of a complex number is a complex number having the same two terms with the sign inbetween changed.

Complex Number	Complex Conjugate
7 + 3i	7 - 3i
3 - 2i	3 + 2i
-2 + 5i	-2 - 5i
i	-i

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Dividing Complex Numbers

1. Change all imaginary numbers to bi form.
2. Write the division problem as a fraction.
3. Rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.
4. Write the answer in the form a + bi.

Example:   Divide

7 + 3i 7 - 3i

7 + 3i 7 - 3i

=

7 + 3i 7 - 3i

·

7 + 3i 7 + 3i

=

(7 + 3i)(7 + 3i) (7 - 3i)(7 + 3i)

=

49 + 21i + 21i + 9i2 49 + 21i - 21i - 9i2

=

49 + 42i + 9(-1) 49 - 9(-1)

=

40 + 42i 49 + 9

=

40 + 42i 58

=

40 58

+

42 58

i

=

20 29

+

21 29

i

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