How do we multiply and divide rational expressions?

Multiplying and Dividing rational expressions are really easy. All you need to remember is to multiply straight across then simplify, simple as that! Like this:

Here you factor the expressions out first than you multiplied straight across. 4x+8 over  x²-25 times x-5 over 5x+10. You take factor each expression and you get 4(x+2) over (x+5)(x-5) times x-5 over 5(x+2). Than you cancel out what the expressions have in common like x+2 and x-5. Multiply straight across and viola! the answer is 4 over 5(x+5).

Dividing rational expressions includes a different step. In order to divide rational expressions you must KCF, which stands for Keep Change Flip. You must keep the first expression change the division sign to a multiplication sign than flip the second expression. Like this:

Here you keep the 2x²y over 3yz and change the division sign to a multiplication sign and flip the expression to 16xd²   over 9y²z. Than you cross out what is in common to simplify the expression and multiply straight across, and there is your answer!




Try This:



Sources:

• http://virtualnerd.com/images/Tutorial/Alg1_8_1_4-diagram_thumb-lg.png
• http://www.virtualnerd.com/images/Tutorial/Alg1_08_01_0019-diagram_thumb-lg.png
• http://hotmath.com/help/solutions/genericalg1/7/2/RationalExpressionsandRationalEquations/genericalg1_7_2_RationalExpressionsandRationalEquations_25_605/image062.gif

How do we simplify expressions with rational exponents?

Rational exponents are fractions, for example like this:

The numerator is the number of times the base is to be multiplied. The denominator is the root being taken.
You'll know your expression is simplified when:

1. The exponent isn't negative: You must flip the exponent and put it in the denominator and make it positive.
2. When the index are really simplified: if you can simplify the fraction exponent, do it! Its much easier!

Lets try a Problem:

Here 27 is raised to the 2/3rd power. The denominator is the root, so 27 to the cube root is 3. Than you square 3 and your final answer is 9. Viola! Simple as that.

Now Try this problem on your own:
What is 25/9 to the 1/2 Power?






Sources:

• http://www.solving-math-problems.com/image-files/math_rad_frac_exp.png
• https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcQ5GY6vvbHS0glig6_EY2DjtvWd45kOcuOvBBgbKuY9xhD06OR9
• http://www.mathops.com/free/a1rr010.php

How do we rationalize a denominator?

Fractions are quite complicated. The denominator of a fractions is only limited to real integers. Having a radical as a denominator makes the fraction untrue, because it will be an irrational fraction. Therefore, we need to rationalize the denominator. Rationalizing the denominator means to simplify the fraction by getting rid of the radical sign.

How do you rationalize?
Easy! You rationalize the denominator by multiplying the denominator by its conjugate. A conjugate is when you change the sign of the binomial to get rid of the radical. For example:

Here, we need to rationalize the denominator therefore we multiply the√ ̅11 +2 by √ ̅11 -2. Remember whatever you multiply the denominator with you multiply the numerator as well! We multiply 3 by √ ̅11  -2 and viola! Your final answer is 3√ ̅ 11-6/7.


Try This:



Sources:

• http://images.planetmath.org/cache/objects/8052/js/img1.png
• http://tutorial.math.lamar.edu/Classes/Alg/Radicals_files/eq0085M.gif

Factoring by Grouping

Factoring is when the common numbers of a expression or equation is divided out to simplify the expression or equation. Factoring is helpful when solving inequalities, linear inequalities, finding the roots of a quadratic equation and more. However, the easiest way to factor is to factor by grouping.

For example:

In this example, you group 2wx+10w and 7x+35. Than you find the Greatest Common Factor for both of the groupings. You should than get 2w(x+5)+7(x+5). Than you group the like terms which are (x+5) and (2w+7) and viola you are done! The answer is (x+5)(2w+7).

Now, try factoring this: 2x^2-4x+3x-6





Sources:

• http://www.coolmath.com/algebra/04-factoring/images/07-factoring-16.gif

Quadratic Inequalities

Similar to Linear Inequalities, solving them is quite easy. The easiest way to solve them is by graphing the inequality.

For Example In this image above
The quadratic inequality is "Y is less than 2x^2+4x-8" Because it is a less than sign the quadratic is dashed. Than you pick a test point. Above the test point picked is (0,0). You plug in (0,0) in the inequality and if the statement isn't true you shade the outside, if it is true you shade in the parabola.

And that is how you solve Quadratic Inequalities!

Try this:
y > x^2+7x+10




Sources:

• http://algebratesthelper.com/wp-content/uploads/2011/02/graphing-quadratic-inequalities.jpg

Simplifying Imaginary Numbers, yes, Imaginary!

What are Imaginary Numbers?

• Imaginary Numbers are complex numbers, which is in the form of a+bi. A and B are real numbers and i is the square root of -1.

Simplifying Imaginary Numbers are easy. Since the square root of i is -1, simplifying negative square roots are easy. 

For example:

The square root of -9.

To simplify the square root of -9 you must find the square root of -1, which is i.

Than, you find the square root of 9 because the square root of -1 is already found.

The square root of 9 is 3, so finally your answer is 3i.

And that is how you simplify imaginary numbers.

Solving Quadratic Equations

Quadratic equations are equations where the highest power is 2. Quadratic equations look like this: ax ² + bx + c. Sometimes the b value is missing, other times the c value is missing. Quadratic equations can be solved two different ways by factoring or using the quadratic formula. Solving a quadratic equation means finding the  roots of the equation. Roots are the solutions which may satisfy the equation. 

Factoring:
Factoring a quadratic equation is only possible if the equation is factor-able. For example: 

Here the equation is x ² + x - 42. It can be factored to (x+7)(x-6). Than you set them to zero and solve for x. The solutions for x are -7 and 6. 

The Quadratic Formula:
The quadratic formula works for every quadratic equation even the ones that can be factored. The quadratic formula is  . In order to solve the equation using the formula you must substitute the numbers in the equation. For example:

Here the equation is x ² + 2x - 3. The a is 1, b is 2 and c is -3. Here all the numbers are plugged in and solved for. Reminder, always find both solutions of the equation by subtracting and adding the numbers properly using the order of operations. The roots for this solution is 1 and -3.

Question:
What are the roots for this quadratic equation? x² + x – 4 = 0





Sources:

• http://www.google.com/url?source=imglanding&ct=img&q=http://www.gradeamathhelp.com/image-files/solve-by-factoring.jpg&sa=X&ei=pWuMUOOSNcr20gHqpID4Dw&ved=0CAkQ8wc&usg=AFQjCNE7eD9GbzXwexGHnCXHnZOttkkdnw
• http://www.google.com/url?source=imglanding&ct=img&q=http://www.mathwarehouse.com/quadratic/images/formula_solution_-3_and_1.gif&sa=X&ei=Bm-MUPCmD4f50gHJtoHgCg&ved=0CAkQ8wc&usg=AFQjCNHsIqk4Z6puR4MNs44agEguLJJh_A

The Purpose of Flipping Inequality Symbols when Multiplying by a Negative Number and Solving for Absolute Value Inequalities!

Many of you solved absolute value inequality problems and regular inequality problems, but never understood why the inequality symbols must be flipped. Well...

Absolute Value Inequalities

The definition of absolute value is, the distance a number is from zero. Therefore, a negative number and a positive number have the same distance from zero. For example:

As you notice, -4 and 4 has the same absolute value. When solving for absolute value inequalities the sign is flipped because of the number line. On the number line both negative and positive numbers count because they both have a distance to Zero. 

Regular Inequality Problems:

Inequality problems on a number line show what values the variable is greater or less than. Therefore, when multiplying by a negative number the sign flips because it is actually less than what the variable can be. On a number line, the inequality sign must be flipped to satisfy the inequality. For Example,

As you can see, the flipping of the sign just makes the inequality true because on a number line a negative number is less than the variable or number used in the problem.

 

Question:

Solve for x: -3|x-9|>27. How many times did you flip the inequality sign?

Sources: 

• http://neaportal.k12.ar.us/wp-content/uploads/2010/08/sei2a14_solve-and-graph-absolute-value-equations-and-inequalities.jpg
• https://d3pof0dg6ipqi1.cloudfront.net/learning_preview/10913/image/large_hqdefault.jpg