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# 10.1 Radical Expressions and Graphs - PowerPoint PPT Presentation

10.1 Radical Expressions and Graphs. is the positive square root of a, and is the negative square root of a because If a is a positive number that is not a perfect square then the square root of a is irrational. If a is a negative number then square root of a is not a real number.

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## PowerPoint Slideshow about ' 10.1 Radical Expressions and Graphs' - livvy

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• is the positive square root of a, andis the negative square root of a because
• If a is a positive number that is not a perfect square then the square root of a is irrational.
• If a is a negative number then square root of a is not a real number.
• For any real number a:
• The nth root of a: is the nth root of a. It is a number whose nth power equals a, so:
• n is the index or order of the radical
• Example:
• The nth root of nth powers:
• If n is even, then
• If n is odd, then
• The nth root of a negative number:
• If n is even, then the nth root is not a real number
• If n is odd, then the nth root is negative
10.2 Rational Exponents
• Definition:
• All exponent rules apply to rational exponents.
10.2 Rational Exponents
• Tempting but incorrect simplifications:
• Review: Expressions vs. Equations:
• Expressions
• No equal sign
• Simplify (don’t solve)
• Cancel factors of the entire top and bottom of a fraction
• Equations
• Equal sign
• Solve (don’t simplify)
• Get variable by itself on one side of the equation by multiplying/adding the same thing on both sides
• Simplified Form of a Radical:
• All radicals that can be reduced are reduced:
• There are no fractions under the radical.
• There are no radicals in the denominator
• Exponents under the radical have no common factor with the index of the radical
• Pythagorean Theorem: In a right triangle, with the hypotenuse of length c and legs of lengths a and b, it follows that c2 = a2 + b2
• Pythagorean triples (integer triples that satisfy the Pythagorean theorem): {3, 4, 5}, {5, 12, 13}, {8, 15, 17}

c

a

90

b

• Distance Formula: The distance between 2 points (x1, y1) and (x2,y2) is given by the formula (from the Pythagorean theorem):
• Example:
• Like Radicals (similar to “like terms”) are terms that have multiples of the same root of the same number. Only like radicals can be combined.
• Tempting but incorrect simplifications:
10.5 Multiplying and Dividing Radical Expressions
• Use FOIL to multiply binomials involving radical expressions
• Example:
10.5 Multiplying and Dividing Radical Expressions
• Examples of Rationalizing the Denominator:
10.5 Multiplying and Dividing Radical Expressions
• Using special product rule with radicals:
10.5 Multiplying and Dividing Radical Expressions
• Using special product rule for simplifying a radical expression:
• Squaring property of equality: If both sides of an equation are squared, the original solution(s) of the equation still work – plus you may add some new solutions.
• Example:
• Solving an equation with radicals:
• Isolate the radical (or at least one of the radicals if there are more than one).
• Square both sides
• Combine like terms
• Repeat steps 1-3 until no radicals are remaining
• Solve the equation
• Check all solutions with the original equation (some may not work)