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

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.

Algebra I – Quadratics

Quadratic equations: Learning how to solve equations is very important subject in algebra. 
Graphing a parabola: Plot points to draw your very first parabola
Graphing quadratics: Every quadratic graph boil down to these 3 numbers
Factoring quadratics: An extremely useful trick for solving some equations
The quadratic formula: A method for solving ANY quadratic equation!
Discriminants and roots: A quick way to tell how many solutions a quadratic has

In this lesson, we will learn how to solve quadratics equations with 2 degrees that can be written in the standard form 〖ax〗^2+bx+c=0 where a,b, and c are real numbers and a≠0. A solution of a quadratic equation in standard form is called a root that can have two real solutions, one real solution, or no real solution.

Algebra I – Polynomials

In this article we will learn: 
- What are polynomials? In this lesson we will learn about a common type of algebraic expression
- Adding polynomials: computing sums and differences of polynomials
- Multiplying monomials: Multiply and divide polynomials with one term
- Multiplying binomials: Multiply binomials together using FOIL (first, outside, inside, and last)
- Multiplying polynomials: How to multiply polynomials with many terms
- Difference of squares: A neat trick for simplifying certain binomials

Polynomials are variables raised to non-negative integer powers, multiplied by coefficients, then added together.

The degree is the biggest exponent in a polynomial. Polynomials with degree 1 are called Linear Polynomial, polynomials with degree 2 are called Quadratic Polynomial, polynomials with degree 3 are called Cubic Polynomial and polynomials with degree 4 are called Quartic Polynomial.

Leading coefficient is the coefficient for the highest power and a polynomial’s Constant Term is the coefficient of the zeroth power.

Algebra I – Lines

In this lesson we will learn:
- Graphing an equation: turn an equation into your very first graph!
- Slope-intercept form: turning an equation into a line, and vice versa.
- Point-slope form: a formula for when you know the slope and coordinates.
- Reading graphs: forget equations, let's just look at the graphs!
- Intercepts: figuring out where lines cross the x- and y-axes
- Solving for intersections: given two lines, can you determine where they cross?
- Graphing absolute value: exploring a graph that's not a line
- Inequalities (2 variables): Simplify inequalities with two variables and test solutions
- Graphing inequalities: Graph inequalities on the coordinate plane

The graph for y= ?x+ ? equation is always a straight line, the value “?” is determine the line’s orientation.
At the y-intercept, x equals zero and at x-intercept y equals zero.
y=mx+b: m is the coefficient for x and it will always be the slope of the line, b is the y-intercept and this equation we can called Slope-intercept form.

Algebra I – Slope

In this lesson we will learn:
- Introduction to Slope: A number that tells you the steepness of a line
- Slope formula: How to calculate the slope between any two points
- Negative slopes: discover what it means when slope is negative
- Zero slope: what are the slopes of horizontal and vertical lines?
- Find the slope of any line: two steps that will always give you the correct slope
- Parallel slopes: Explore why parallel lines have the same slope
- Perpendicular slopes: Discover how slopes of perpendicular lines relate

Slope is the vertical distance divided by the horizontal distance.

Negative slopes moving left to right and they go down while, positive slopes moving left to right and they go up.

Horizontal lines have a slope of zero and vertical lines have a slope that’s undefined.

Algebra I – Coordinates - The coordinate plane | Quadrants | Finding a midpoint

The horizontal number line is called the x-axis, and the vertical number line is called the y-axis in rectangular coordinate system which intersect at a right angle, first number is x-coordinate and second number is y-coordinate. The two number lines define a flat surface is called a plane, and each point on this plane is associated with an ordered pair of real numbers (x, y) with origin point (0, 0).