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10.1 Radical Expressions and Graphs

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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

(0, 0)

- 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

- Expressions

- Product rule for radicals:
- Quotient rule for radicals:

- 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):

- We can add or subtract radicals using the distributive property.
- 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: