Chapter+11

//"Chapter 11 is algebra's way of apologizing for chapter 10"// //-Kayla// =** 11-1 **=
 * ** Simplify radical expressions using the Product Property of Square Roots. **
 * ** Simplify radical expressions using the Quotient Property of Square Roots. **

√180 √2*2*3*3*5  2² * 3² * 5  2*3 √5  6 √5
 * Example: **

The square root of **A** times the square root of **B** is equal to the square root of **AB ( **√ab = √a * √b**)** .
 * Steps: **
 * 1) Prime Factorization
 * 2) Combine ---> x²
 * 3) Square parts you can
 * 4) Simplify

If whatever is **outside** of your square root symbol has an **odd** exponent, it **needs** to be put in the absolute value symbol (No exponent- assume it is to the power of 1)

// When your multiplying exponents you are really just adding them (x² * x^6 = x^8) // //.// **Fraction Problem Steps:** 1. Write problem 2. Multiply by a fraction equal to one (1/1) .... a. If you start with (-) change it to (+) .... b. If you start with (+) change it to (-) 3. Multiply numerators 4. Multiply denominators 5. Simplify numerator/denominator 6. Reduce fraction

= 11-2 =
 * Add and subtract radical expressions
 * Multiply radical expressions

**Expressions with Like Radicands:** 1. Distributive property 2. Simplify

**Expressions with Unlike Radicands:** 1. Prime factorization of numbers inside square root 2. Combine x² 3. Take square root of what you can 4. Combine/Simplify

**Multiply Radical Expressions:** 1. FOIL 2. Multiply 3. Prime factorization of numbers inside square root 4. Combine x² 5. Take square root of what you can 6. Simplify 7. Combine like terms

= 11-3 =
 * Solve radical expressions
 * Solve radical equations with extraneous solutions

**Variable on one side:** 1. Square Each Side 2. Simplify

√3x + 12 = 5 3x + 12 = 25 3x = 13 x = 4.33

**Variable on Each Side:** 1. Square both sides 2. Simplify 3. Subtract/add to equal 0 4. Factor 5. Solution set 6. Only one number will work

//** If the number is negative it is always an extraneous solution (no solution) **//

= 11-4 =
 * Solve problems by using the Pythagorean theorem
 * Determine whether a triangle is a right triangle

Will only work with **right triangles!** **a^2 + b^2 = c^2** c^2 - a^2 = b^2 c^2 - b^2 = a^2

**Pythagorean Triples:** Whole numbers that satisfy the Pythagorean Theorem . 3, 4, 5  . 6, 8, 10  9, 12, 15  infinitely more...

A and B can equal either leg
 * C will always equal the hypotenuse (biggest number) **

= 11-5 =
 * Find the distance between two points on the coordinate plane.
 * Find a point that is a given distance from a second point in a plane

Distance formula: used to find distance between 2 points on a graph

-When looking for distance your answer can only be positive -When your looking for a missing x or y value it is ok to have more than one answer

__Solving Distance Formula Backwards:__
 * Wrote problem
 * Put in numbers (replaced variables)
 * Simplified
 * Squared both sides
 * Made one side equal to zero
 * Factored
 * Used zero product of multiplication
 * Check your work (plug in answer)

= 11-6 =
 * Determine whether two triangles are similar
 * Find the unknown measures of sides of two similar triangles

Similar- same shape; different size Corresponding- same spot on a different figure

// Angles are the same therefore the triangles are similar // //.// //.// = 11-7 =
 * Define the sine, cosine, and tangent ratios
 * Use trigonometric ratios to solve right triangles

__ SOH CAH TOA: __ // Sin Opposite Hypotenuse // // Cosine Adjacent hypotenuse // // Tangent Opposite Adjacent //

** . **  **Hypotenuse** - Line across from the right angle **Adjacent-** Line closest to theda never the hypotenuse **Opposite** - Line opposite from theda
 * ** It is a ratio that describes a relationship between a side and angle of a triangle. **
 * ** The decimal equivalent to the ratio of one side of a right triangle to another. **

Sine- (sin) Cosine- (cos) Tangent- (tan) . **SohCahToa-** **S** ine= **O** pposite / **H** ypotonuse **C** osine= **A** djacent / **H** ypotonuse **T** angent= **O** pposite / **A** djacent . Opposite is opposite to the theda Adjacent is right under the theda . ** Theda is oval with line through it (the angle we are finding) ** **How to solve-** 1. find sides (opposite, adjacent, hypotenuse) 2. determine if it is sine, cosine, or tangent 3. find angles and stuff 4. turn sine, cosine, or tangent into a number 5. get x alone (answer)

** If x is on the top you multiply, if x is on the bottom you divide **