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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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Presentation Transcript

• 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

• Definition:

• All exponent rules apply to rational exponents.

• Tempting but incorrect simplifications:

• Examples:

• 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

• Example:

• Example:

• 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:

• Use FOIL to multiply binomials involving radical expressions

• Example:

• Examples of Rationalizing the Denominator:

• Using special product rule with radicals:

• 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)

• Example:Add 1 to both sides:Square both sides:Subtract 3x + 7:So x = -2 and x = 3, but only x = 3 makes the original equation equal.

• Definition:

• Complex Number: a number of the form a + bi where a and b are real numbers

• Adding/subtracting: add (or subtract) the real parts and the imaginary parts

• Multiplying: use FOIL

• Examples:

• Complex Conjugate of a + bi: a – bimultiplying by the conjugate:

• The conjugate can be used to do division(similar to rationalizing the denominator)

• Dividing by a complex number: