2. CHAPTER R: Basic Concepts of Algebra
R.1 The Real-Number System
R.2 Integer Exponents, Scientific Notation, and Order of
Operations
R.3 Addition, Subtraction, and Multiplication of Polynomials
R.4 Factoring
R.5 The Basics of Equation Solving
R.6 Rational Expressions
R.7 Radical Notation and Rational Exponents
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3. R.7 Radical Notation and Rational Exponents
Simplify radical expressions.
Rationalize denominators or numerators in rational expressions.
Convert between exponential and radical notation.
Simplify expressions with rational exponents.
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Notation
A number c is said to be a square root of a if c2 = a. Similarly, c is a cube root of a if c3 = a.
nth Root
A number c is said to be an nth root of a if cn = a.
The symbol denotes the nth root of a. The symbol is called a radical. The number n is called the index.
Any positive number has two square roots, one positive and one negative. For any even index, a positive number has two real-number roots. The positive root is called the principal root.
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Examples
Simplify each of the following:
a) = 7, because 72 = 49.
b) = 7, because 72 = 49 and
c) because
d) because (4)3 = 64.
e) is not a real number.
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6. Properties of Radicals
Let a and b be any real numbers or expressions for which the given roots exist. For any natural numbers m and n (n  1):
1. If n is even,
2. If n is odd, 3. 4. 5.
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Examples a) b) c) d) e) f) g) h)
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Another Example
Perform the operation.
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9. The Pythagorean Theorem
The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse:
a2 + b2 = c2. b c a
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Example
Juanita paddled her canoe across a river 525 feet wide. A strong current carried her canoe 810 feet downstream as she paddled. Find the distance Juanita actually paddled, to the nearest foot.
Solution:
525 ft
810 ft c
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11. Rationalizing Denominators or Numerators
Rationalizing the denominator (or numerator) is done by multiplying by 1 in such a way as to obtain a perfect nth power.
Example
Rationalize the denominator.
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12. Rationalizing Denominators or Numerators
Conjugates
Pairs of expressions of the form
are called conjugates. The product of such a pair contains no radicals and can be used to rationalize a denominator or numerator.
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13. Rationalizing Denominators or Numerators
Example
Rationalize the numerator.
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Rational Exponents
For any real number a and any natural numbers m and n, for which exists,
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Examples
Convert to radical notation and, if possible, simplify. a) b) c)
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More Examples
Convert each to exponential notation. a) b)
Simplify.
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