Algebra, Chapter 10
Simplest Radical Form
1. No radicands have perfect square factors other than one 2. No radicands contain fractions 3. No radicals appear in denominator
Rules of using imaginary numbers
1. The end product should have no i's in the denominator 2. The end product should have the negative on top if its a fraction 3. The powers of i are always in a pattern of four 4. Never leave exponents on i 5. Two negative radicals cannot be multiplied until they are simplified, with no negative under the square root 6. Only use imaginary numbers when asked to
The powers of i are in what kind of pattern?
A pattern of four; i^2=-1, i^3=-i, i^4=1, i^5=i, repeat. To find what a bigger piwer of i equals, divide by four. If the remainder is .25, =i. If the remainder is .5, =-1. If the remainder is .75, =-i. If it evenly goes into four, =1.
How to find if triangles are similar
Add up the measures of the angles and subtract that from 180. Dothe same to the other triangle. If both triangles have same angle measures, they are similar.
True or False: It matters what order you subtract the coordinates points
False
How to Simplify Square Roots
Find the largest perfect square that goes into the radicand. Separate the numbers, with the perfect square first. Solve the perfect square and leave it next to the square.
Rationalizing the denominator
If a fraction is under the radical sign or a radical is in the denominator, times it by the square root of the bottom number. Simplify. When a conjugate is in the denominator, times it by its conjugate (same numbers, opposite signs). Simplify.
Adding and subtracting radical expressions
Radical expressions in which radicands are alike can be added or subtracted in the same way that monomials are added or subtracted. Numbers outside the radicals are added together.
How to solve a negative square root with imaginary numbers
Separate the square root, with the square root of negative one first, then the positive perfect square, then the last square root. Since i= √-1, solve the perfect square and add i behind it.
Exponents are perfect squares when...
They are even
Pythagorean Triple
Three whole numbers that satisfy the Pythagorean Theorem e.i. 3,4,5; 3^2+4^2=5^2
Solving radical equations
To solve, first isolate the radical on one side of the equation. Then, square each side to eliminate the radical.
True or False: You MUST check the distance formula for extraneous solutions
True
True or False: you can never leave a radical in the denominator
True
True or False: you cannot multiply two negative radicals together until they are simplified with no negative under the square.
True
Extraneous Solution
a solution derived from an equation that is not a solution of the original equation (sometimes happens when you square each side of an equation). Therefore you must check all solutions in the original equation when solving radical expressions.
Radical Expression
an expression that contains a square root
Pythagorean Theorem
a²+b²=c² a+b= legs c= hypotenuse *Only works on roght triangles
Conjugates
binomials in the form p √q + r √s, 2 binomials that have a radical in them and are identical
Distance Formula
d=√(x₂-x₁)²+(y₂-y₁)², where xs and ys are coordinate points
radical equations
equations like 3= √x/4 that contain radicals with variables in the radicand.
Radicand
expression under the radical sign
When solving radical equations, you can...
graph, factor, complete the square or use quadratic formula
Imaginary Numbers
i= √-1, only use imaginary numbers when questions ask for them
Product Property of Square Roots
if a≥0 and b≥0 then √ab = √a×√b
A radicand is in simplest form when...
it contains no perfect square greater than one
When multiplying roots...
multiply the numbers outside root together and inside root together.
If a checked solution doesn't work it's...
no solution
Simplifying a negative square is possible...
only with imaginary numbers
If an even exponent on a variable comes out odd after the square...
put absolute value around it
Congruent Triangles
same shape figure, same size figure
Similar Triangles
same shape triangle or figure, but not necessarily the same size
≅
similar sign
Sign in front of radical sign...
tells what # (positive or negative) is wanted
If two triangles are similar, then...
the measurements of their corresponding sides are proportional and the measurements of corresponding angles are similar
True or False: fractions cannot be under the radical
true
There is always ___ answers to a square root equation, including the Pythagorean Theorem- a ___ answer and a ___ answer.
two, negative, positive (but a triabgle lenght cannot be negative)
