Algebra and Geometry Review – Exercise 10

LESSON 1: Rational exponents (Unit fraction exponents and whole number bases)

Evaluate: 27^(1/3)

256^(1/4)

EXPLANATION: We can use the following to evaluate the exponential expressions

a^(1/n)=√(n&a)

We have 27^(1/3)=∛27. So, we need to find the cube root of 27

Checking some positive integers, we see that 3 is the cube root of 27.

1^3=1*1*1=1

2^3=2*2*2=8

3^3=3*3*3=27

So, 27^(1/3) = 3

We have 256^(1/4)=∜256. So, we need to find the fourth root of 256

Checking some positive integers, we see that 4 is the fourth root of 256

1^4=1*1*1*1=1

2^4=2*2*2*2=16

3^4=3*3*3*3=81

4^4=4*4*4*4=256

So, 256^(1/4) = 4

ANSWER: 27^(1/3) = 3, 256^(1/4) = 4

Algebra and Geometry Review – Exercise 09

LESSON 1: Adding rational expressions with common denominators and monomial numerators

Subtract and Simplify: 8/(b+3)-1/(b+3)

EXPLANATION: 
Note that 8/(b+3) and 1/(b+3) have the same denominator b+3
So, we subtract with the numerators and the denominator stays the same.
8/(b+3)-1/(b+3)=(8-1)/(b+3)=7/(b+3)

ANSWER: 7/(b+3)

LESSON 2: Introduction to simplifying a radical expression with an even exponent

Simplify and Assume that the variable represents a positive real number: √(y^10 )

EXPLANATION: By definition, √(y^10 ) is a positive number whose square equals y^10
Note that 〖(y^5)〗^2=y^5*y^5=y^(5+5)=y^10 
Therefore, the square of y^5 is y^10
And in this problem, we are assuming y is positive, so y^5 is also positive
So, we have the following: √(y^10 )=y^5
Or √(y^10 )=y^(10÷2)=y^5

ANSWER: y^5

LESSON 3: Using distribution with double negation and combining like terms to simplify (Multivariate)

Simplify -3x-5(-5y+2x)-4y

EXPLANATION: 
-3x-5(-5y+2x)-4y
=-3x+25y-10x-4y  Using distributive property to remove parentheses
=-3x+(-10x)+25y+(-4y) Using commutative property to rearrange terms
=-13x+21y  Combining like terms

ANSWER: -13x+21y  

LESSON 4: Rewriting an algebraic expression without a negative exponent

Rewrite the expression without using a negative exponent and Simplify as much as possible
1/(2p^(-2) )

EXPLANATION: For any nonzero number a and any whole number n, we have the following
Rule 1: a^(-n)=1/a^n Move a^(-n) to the denominator and make the exponent positive
Rule 2:  1/a^(-n) =a^n Move a^(-n) to the numerator and make the exponent positive
We need to rewrite 1/(2p^(-2) ) without a negative exponent
We use Rule 2 and move p^(-2) to the numerator, making the exponent positive
1/(2p^(-2) )=p^2/2

ANSWER: p^2/2

LESSON 5: Multiplying binomials with negative coefficients

Multiply and Simplify: (-8y+5)(5y-2)

EXPLANATION: We multiply as follows
(-8y+5)(5y-2)=-40y^2+16y+25y-10 Using FOIL
=-40y^2+41y-10 Simplifying

ANSWER: -40y^2+41y-10

Algebra and Geometry Review – Exercise 08

LESSON 1: Degree and leading coefficient of a univariate polynomial

What are the leading coefficient and degree of the polynomial?
3-3x^2+6x

EXPLANATION: We first rewrite this polynomial in standard form. 
We rearrange the terms so that the exponents on the variable decrease from left to right.
3-3x^2+6x=-3x^2+6x+3 standard form
The leading term is the first term when the polynomial is in standard form.
So, for this polynomial, the leading term is -3x^2

Leading coefficient
The coefficient of a term is the number multiplying the variable,
when there is no variable, the coefficient is the term itself
The leading coefficient is the coefficient of the leading term.
For our polynomial, the leading term is -3x^2. So, the leading coefficient is -3.

Degree
The degree of a term is the exponent of its variable,
when there is no variable, the degree is zero
The degree of a polynomial is the degree of its leading term
For our polynomial, the leading term is -3x^2. So, the degree of the polynomial is 2.

Algebra and Geometry Review – Exercise 07

LESSON 1: Product rule with positive exponents (Multivariate)

Multiply and simplify your answer as much as possible:  5v^4 w^2*3w^8*2v

EXPLANATION: Grouping similar factors, we get the following
5v^4 w^2*3w^8*2v=(5*3*2) v^4 vw^2 w^8
Then, we simplify using the properties of exponents
30v^(4+1) w^(8+2)=30v^5 w^10

ANSWER: 30v^5 w^10

LESSON 2: Introduction to square root multiplication

Simplify √3*√7

EXPLANATION: We'll use the following property of square roots to simplify our expression

for any nonnegative numbers a and b: √a*√b=√ab 

We simplify as follows: 

√3*√7= √(3*7)  Using the product property of square roots

=√21 Multiplying under the square root sign

The number under the square root 21 has no perfect square factors other than 1

Therefore, √21  is in simplified radical form and is our answer.

ANSWER: √21

Algebra and Geometry Review – Exercise 06.1

LESSON 1: Factoring a quadratic with leading coefficient greater than 1 (Problem type 1)

Factor: 2x^2+17x+15

EXPLANATION: Method 1 - We will factor using a method often called Trial and Error. 

The coefficient of x^2 is 2

Using whole numbers, only 1 and 2 have a product of 2

So we'll look for integers n and m that satisfy the following.

2x^2+17x+15 = (1x+m)(2x+n) 

Using FOIL to expand the right-hand side of this equation, we get the following

nm=15 and n+2m=17

We'll find all integers m and n such that mn=15

Then we'll check to see if n+2m=17

Algebra and Geometry Review – Exercise 06

QUESTION 1: Introduction to the GCF of two monomials

Find the greatest common factor of: 12x^2 and 8x^4

EXPLANATION:

The GCF of 12 and 8 is 4

The GCF of x^2 and x^4 is x^2

So, the GCF of 12x^2 and 8x^4 is 4x^2

ANSWER: 4x^2

QUESTION 2: Factoring a difference of squares in one variable

                Factor: u^2 – 4

EXPLANATION: Here is the factoring formula for the difference of squares

                A^2 – B^2 = (A + B)(A - B)

                We can use this for the current problem

                u^2 – 4 = u^2 – 2^2         Writing as A^2 – B^2, with A=u and B=2

                = (u + 2)(u - 2)    Using the difference of squares formula

ANSWER: (u + 2)(u - 2)

Algebra and Geometry Review – Exercise 05

QUESTION 1: Multiplying a univariate polynomial by a monomial with a positive coefficient

Use the distributive property to remove the parentheses and Simplify your answer as much as possible:

                10b^2(4b + 2b^5)

EXPLANATION: We use the distributive property and then simplify, as follows

                10b^2(4b + 2b^5) = 10b^2 * 4b + 10b^2 *2b^5   Distributing 10b^2 across the parentheses

                = (10*4)b^2*b + (10*2)b^2*b^5                Grouping similar factors

                = (10*4)b^3 + (10*2)b^7               Using the product rule for exponents

                = 40b^3 + 20b^7

ANSWER: 40b^3 + 20b^7

QUESTION 2: Square roots of perfect squares with signs

Evaluate the following and write "Not a real number" if applicable.

                Minus root 25: -sqrt(25)

                Root minus 36: sqrt(-36)

EXPLANATION: In these problems, we must deal with negative signs and square roots.

                We have that sqrt(a) is a real number only if a is positive or 0

                If a is negative, then sqrt(a) is not a real number

                We will use this fact in the current problem

                Note: sqrt(25) = 5, then mean that -sqrt(25) = -5

               

                Sqrt(-36) : If a is negative, then sqrt(a) is not a real number. So, Sqrt(-36) is not a real number

ANSWER:

                -sqrt(25) = -5

                sqrt(-36) = Not a real number

Algebra and Geometry Review – Exercise 04

QUESTION 1: Factoring a perfect square trinomial with leading coefficient 1

                Factor: y^2 - 12x + 36

               

EXPLANATION: When factoring a polynomial, the following formulas are sometimes useful

1.       a^2 + 2ab + b^2 = (a + b)^2

2.        a^2 - 2ab + b^2 = (a - b)^2

We can use formula 2. to factor the given perfect square polynomial.

y^2 - 12x + 36 = y^2 – 2(y)(6) + 6^2

                = (y – 6)^2

ANSWER: (y – 6)^2

QUESTION 2: Simplifying a ratio of multivariate monomials

                Simplify: 25xy / 35yz

EXPLANATION: We simplify 25xy / 35yz as follows

                25xy / 35yz = 5xy / 7yz                   Canceling the common factor 5

                = 5x / 7z               Canceling the common factor y

ANSWER: 5x / 7z

Algebra and Geometry Review – Exercise 03

QUESTION 1: Evaluating an expression with a negative exponent

Rewrite the following without an exponent: 8^-2

EXPLANATION:

For any nonzero number a and any whole number n, we have the following.

Rule 1:                  a^-n = 1/a^n

Move a^-n to the denominator and change -n to n

Rule 2:                  1/a^-n = a^n

Move a^-n to the numerator and change -n to n

We need to rewrite 8^-2 without an exponent:

To do this, we first use Rule 1 and move 8^-2 to the denominator, making the exponent positive and evaluate

8^-2 = 1/8^2       using Rule 1

= 1/64                   Since 8^2 = 64

ANSWER: 1/64

Algebra and Geometry Review – Exercise 02

QUESTION 1: introduction to the quotient rule of exponents

Simplify y^7/y^4

EXPLANATION: The exponents tell us how many y's to multiply

                y^7/y^4 = Y*y*y*y*y*y*y / y*y*y*y

                = y*y*y / 1          Canceling gives us the following

                = y^3

ANSWER: y^3

QUESTION 2: Multiplying binomials with leading coefficients of 1

Multiply and Simplify your answer: (u-2)(u+7)

EXPLANATION:

                We want to remove the parentheses from the product (u-2)(u+7)

                We first multiply each term in the first factor (u-2) by each term in the second factor (u+7) using FOIL (First, Outer, Inner, Last).

                F:            Multiply the two First terms: u*u = u^2

                O:           Multiply the two Outside terms: u*7 = u7

                I :            Multiply the two Inside terms: -2*u = -2u

                L:             Multiply the two Last terms: -2*7 = -14

               

The product (u-2)(u+7) is then equal to the sum of these terms:

                (u-2)(u+7) = u^2 + 7u + - 2u – 14

                = u^2 + 5u -14

ANSWER: u^2 + 5u -14

Algebra and Geometry Review – Exercise 01

QUESTION 1: Using distribution and combining like terms to simplify: 3(y + 5) - 6y

EXPLANATION

                3(y + 5) - 6y = 3y + 15 – 6y            Using the distributive property to remove parentheses

                = 3y + 15 + (- 6y)               Writing subtraction as addition of a negative

                = 3y + (- 6y) +15                Using the commutative property to rearrange terms

                = - 3y + 15           Combining like terms

ANSWER: -3y + 15

QUESTION 2: Use the distributive property to remove the parentheses (-2 + 4x + 4v) (-7)

EXPLANATION: We use the distributive property as follows

                (-2 + 4x + 4v) (-7) = (-2)(-7) + (4x)(-7) + (4v)(-7)

                = 14 + (-28x) + (-28v)

                = 14 - 28x – 28v

ANSWER: 14 - 28x – 28v

Sensitivity Analysis | Statistics

This lesson we learn:
- Sensitivity and specificity: Quantifying whether tests for diseases are any good
- Predictive value: If you test positive/negative, what are your chances?
- Cutoffs and ROCs: Sensitivity vs. specifity, cutoff selection, and ROC curves
- Prevalence: How rare a disease is, and how it affects PPV and NPV

Positive predictive value (PPV), negative predictive value (NPV)
Receiver operating characteristic (ROC) curves
A rare disease has a prevalence near 0.
The more common a disease, the greater the prevalence.

Tangent lines - Drawing derivatives | Calculus

Tangent lines: Finding the "slope" of a curve
Drawing derivatives: Learn what a derivative is, and draw your own

Tangent lines - Drawing derivatives | Calculus:

Limits | Calculus

In this lesson we learn:
- Limits to infinity: What happens to functions as x gets really, really big?
- Vertical limits: Sometimes y goes off to infinity
- Finite limits: Limits when both x and y stay finite
- One-sided limits: Limits that come from only the left or right side
- Continuity (for real): Dive deeper into continuity, using one-sided limits
- Splitting limits: Tricks for simplifying limits (and when they don't work!)

In this section we will learn about limits whose value is infinity or minus infinity. 
The function f(x) will have a horizontal asymptote at y=L if either of the following are true; limit x go to infinity f(x) = L and limit x go to minus infinity f(x) = L.
The function f(x) will have a vertical asymptote at x = a if we have any of the following limits at x = a; limit x approaching a-negative f(x) = positive/negative infinity, limit x approaching a-positive f(x) = positive/negative infinity, and limit x approaching a f(x) = positive/negative infinity.
We say that the limit of f(x) is L as x approaches a and write this as limit x approaches a f(x) = L, provided we can make f(x) as close to L as we want for all x sufficiently close to a, from both sides, without actually letting x be a.

Piecewise Functions and Compositions

- Piecewise functions: Franken-functions. They're alive!

- Compositions: Functions of functions

Functions | Algebra II

This lesson we learn: 
- Relations: Sets of (x, y) coordinates
- Functions: Relations where each input gives exactly one output
- Functional notation: Explore what f(x) means, and evaluate functions
- Continuity and smoothness: Some functions end and others keep going
- Describing functions: Make your own functions, and live to tell the tale
- Concavity: Math-talk for curvy
- Odd and even functions: Turns out functions can be "odd" and "even" too

A relation is any set or collection of ordered pairs (x, y) in coordinate system which, the set of x-values defines the domain and the set of y-values defines the range.
Special relations where every x-value (input) corresponds to exactly one y-value (output) are called functions.
Continuity – a function is continuous if you can draw it without picking up your pen.
Smoothness – a function is smooth if it’s continuous and doesn’t have any pointy corners.
A function is positive where its outputs are positive

Imaginary and complex numbers

This lesson we learn:
- Imaginary numbers: Their squares are negative!
- Working with imaginaries: Add, subtract, multiply, and divide imaginary numbers
- Complex numbers: Combining the real and the imaginary
- The complex plane: A way to graph complex numbers
- Powers of i: Raising i to positive and negative integer powers

There is no real number that squared results in a negative number, the resolution of this issue by defining the imaginary unit “i” as the square root of -1.
The square root of any negative real number can be written in terms of the imaginary unit or often called imaginary numbers.

Real Numbers - Algebra II

In this lesson we learn:
- Integers: Whole numbers, positives, and negatives
- Rational: Numbers that are quotients of integers
- Irrationals: Numbers that are NOT quotients of integers
- Real numbers: All the rationals and the irrationals

Adding whole numbers always gives you a whole number.
Adding, subtracting and multiplying integers always gives you an integer.
Rational numbers are number that can be written as a quotient of integers.
Adding, subtracting, multiplying and dividing(not divide by zero) rational always gives you a rational.
Irrational numbers are number that can’t be written as a quotient of integers.
Prime factors of squares always come in pairs. 
Rational numbers are number that can be written as a quotient of integers.
Real numbers are all the numbers on the number line includes all the integers and all rational number.
Adding, subtracting, multiplying and dividing(not divide by zero) real numbers always gives you a real number.

Triangle Formulas

In this lesson we learn:
- Law of sines: Relating angles and opposite sides for any triangle
- Triangulation: Real-world application of the law of sines to find distances
- Law of cosines: Relating three sides and an angle for any triangle
- Triangle area (SAS): Find a triangle's area using 2 sides and the angle between

Triangle has sides of lengths a, b, and c opposite the angles A, B, and C then:
a/sin A = b/sin B = c/sin C
Triangle has sides of lengths a, b, and c opposite the angles A, B, and C then:
c^2 = a^2 + b^2 − 2ab cos C

Trig Identities

In this lesson we learn:
- Trig identities: The tools for simplifying trig expressions
- Addition identities: Simplify trig functions of sums
- Negative angle identities: Simplify trig functions of negative angles
- Subtraction identities: Simplify trig functions of differences
- Double angle identities: Trig functions of twice an angle
- Triple angle identities: Trig functions of three times an angle
- Pythagorean identities: Relating squares of trig functions
- Half angle identities: Trig functions of half an angle

sin(2a) = sin(a+a) = sin(a) cos(a) + cos(a) sin (a) = 2 sin(a) cos(a)