Vocabulary Test 4
EX
= 6 Also = 0 (we don't have to write +0)
A. Rational Exponent
A power that is a fraction
C. Extraneous solution
A solution to the derived equation that is not a solution to the original equation. (This is why we must check all solutions to radical equations in the original equation to see if it is indeed a solution or must be excluded as an extraneous solution.) Notice the word 'extra' in the word extraneous! It's an 'extra' answer that doesn't work!
G. Radical
Also called 'root', it's the symbol used to denote the taking of a root.
B. Derived equation
An equation obtained by raising each side of an equation to a power; it may have solutions that do NOT work in the original equation!
A. Radical equation
An equation with a variable under the radical.
L. Radical Function
Any function that can be put in the form where the index (n) is an integer greater than 1. (Basically, it's a function with a variable under the radical).
Similarly
, and are not real numbers!
EX
- = -6
EX
3 is a 4th root of 81 because 34 = 81 = 3
EX
Find the square root of -25. Not a real number
EX
Find the square root of 0. 0 (this is the ONLY number with just 1 square root.)
EX
Find the square root of 36. 6 and -6 because (6)2 and (-6)2 both = 36.
K. Legs
In a right triangle, the legs are the lengths of the sides that are adjacent to the right angle. They are denoted as 'a' and 'b' in the Pythagorean theorem.
A. Division Property of Radicals
Let a and b represent real numbers such that and are both real numbers. Then,
A. Multiplication Property of Radicals
Let a and b represent real numbers such that are both real. Then
B. Magnum
Magnum P.I., a way to remember that with rational exponents the numerator is the power and the denominator is the index.
E. Index
Number in the 'arm-pit' of a radical, tells how many identical factors are necessary for simplification to occur.
I. Pythagorean Theorem
Only works in right triangles, triangles with a 900 angle.
A. Like Radicals
Radicals are considered to be like radical terms if they have same the same index and the same radicand.
B. Addition/Subtraction of Radicals
Radicals can only be added or subtracted if they are like terms. (They must be like radicals).
A. Conjugate
The conjugate of a + b is a - b.
Example
The conjugate of is
B. Simplified Form of a radical
The expression is simplified if ALL the following conditions are met:
J. Hypotenuse
The longest side of a right triangle. It is located across from the right angle and is denoted as 'c' in the Pythagorean theorem.
F. Radicand
The name of the expression under the radical sign.
B. Rationalizing the Denominator
The process of removing a radical from the denominator of a radical expression.
D. Principal Nth root
The radical symbol ( ) is used to denote the principal nth root . The simplified principal Nth root can be positive, negative or zero... see below.
M. Domain of Radical Functions
The set of all inputs for which the function is defined. If the index is odd then the domain is all reals but if the index is even the domain must be restricted to values that make the radicand positive. To restrict the domain, set the radicand 0 and solve for the variable. That solution written in interval notation will be the domain!
B. Special Case Products
These are helpful to know! While they can be found by writing them out and using the distributive property (or F.O.I. L.); having these committed to memory can be helpful and NECESSARY if pursing more math after this class!
B. Principal Square root
When the radical symbol ( ) is used to denote a square root it denotes ONLY the positive square root. This positive square root is called the Principal Square Root.
EX
and are like radical terms.
EX
and are not like radicals.
A. Square Root
b is a square root of a if b2 = a
C. Definition of a nth Root
b is an nth root of a if bn = a
EX
is not simplified because 7 is greater than 3
EX
is not simplified, there is a fraction in the radicand.
EX
is not simplified, there is a radical in the denominator.
EX
is simplified, as it meets all three criteria above.
In
n is called the index, a is called the radicand and the symbol is called the radical.