Roots and Rational Exponents: a summary

Although addition and multiplication are commutative, exponentiation is not: swapping the value in the base with the value in the exponent will produce a different result (unless, of course, they are the same value):

2^3 \ne 3^2

Therefore, two different inverse functions are needed to solve equations that involve exponential expressions:
– roots, to undo exponents
logarithms, to undo bases

Just as there are many versions of the addition function (one for each number you might wish to add), and many versions of the “logarithm” function (each with a different base), there are many versions of the “root” function: one for each exponent value to be undone.

Notation

The symbol for a root is \sqrt{~~~~}, and is referred to as a “radical“.  It consists of a sort of check mark on the left, followed by a horizontal line, called a “vinculum”, that serves as a grouping symbol (like parentheses) to Continue reading Roots and Rational Exponents: a summary

Logarithms

Unlike the two most “friendly” arithmetic operations, addition and multiplication, exponentiation is not commutative. You will get a different result if you swap the value in the base with the one in the exponent (unless, of course, they are the same value):

3^2 \ne 2^3

The most significant impact of this lack of commutativity arises when you need to solve an equation that involves exponentiation: two different inverse functions are needed, one to undo the exponent (a root), and a different one to undo the base (a logarithm).

Just as there are many versions of the addition function (adding 2, adding 5, adding 7.23, etc.), and many versions of the “root” function (square roots, cube roots, etc.),  there are also many versions of the “logarithm” function. Each version has a “base”, which corresponds to the base of its inverse exponential expression.

Inverse Functions: Logarithms & Exponentials

Logarithms are labelled with a number that corresponds to the base of the exponential that they undo. For example, the Continue reading Logarithms

Solving Systems Of Linear Equations

What is a “system” of linear equations?

A “system of linear equations” means two or more linear equations that must all be true at the same time.

When represented symbolically, a system of equations will usually have some sort of grouping symbol to one side of them, such as the curly brace below, which is intended to convey that the set of equations should be considered all at once. For example:

\begin{cases}y=-3x+2\\y=x-6\end{cases}

When graphed, all of the equations in a system will be shown on the same set of axes, so that they can be compared to one another easily:

What is “a solution” to a linear system?

A solution to a system of linear equations is Continue reading Solving Systems Of Linear Equations

3 Ways to “Complete the Square”

I have seen three approaches to “Completing the Square”, as shown below. Each successfully converts a quadratic equation into vertex form.  Which do you prefer, and why?

First Approach

This approach can only be used when you are working with an equation. It moves all terms that are not part of a perfect square to the other side of the equation to get them out of the way:

y~=~2x^2+12x+10

y-10~=~2x^2+12x

\dfrac{y-10}{2}~=~x^2+6x

\dfrac{y-10}{2}+(\frac{6}{2})^2~=~x^2+6x+(\frac{6}{2})^2

\dfrac{y-10}{2}+9~=~x^2+6x+3^2

\dfrac{y-10}{2}+9~=~(x+3)^2

\dfrac{y-10}{2}~=~(x+3)^2-9

y-10~=~2(x+3)^2-18

y~=~2(x+3)^2-8

Second Approach

This approach keeps everything on the same side of the equation, and is therefore suitable for use when you either do not have an equation to work with, or do not wish to “mess up” the rest of the equation with your work to complete the square. It factors the leading coefficient out of all terms before proceeding:

Continue reading 3 Ways to “Complete the Square”

Quadratic Equations: How to Solve Them Algebraically

Adding a squared term to a linear expression, creating a quadratic expression in the process, seems like a relatively small change:

3x+2

x^2+3x+2

Yet, if this new term is part of an equation, the procedures that worked nicely when solving linear equations don’t work so well any more. Investigating what happens in such situations is useful, and leads to some new concepts and procedures.

If a quadratic equation is approached in the same way as a linear equation, it can sometimes be solved quickly:

18=2x^2\\*~\\*9=x^2\\*~\\*\pm3=x

Familiarity with square roots and how to solve linear equations are enough to solve this equation. By getting the variable all by itself on one side, the two possible solutions to the original equation are left on the other.

However, the following equation cannot be solved in the same way:

0=x^2+2x+1\\*~\\*-1=x^2+2x\\*~\\*\dfrac{-1}{x}=x+2~~~~~~~~~(a)\\*~\\*-1=(x)(x+2)~~~(b)

This equation has two “x” terms, and they are not “like terms” since one has the variable to the first power and the other to the second power. If a linear equation approach is used, moving the constant term to the other side, two un-like terms are left on the right… but what to do from here?

Continue reading Quadratic Equations: How to Solve Them Algebraically

Multiplying Polynomials and FFFT!

Most folks learn to multiply increasingly complex quantities gradually over time, starting with constants in elementary school, and eventually continuing on to polynomials in high school.  As the quantities become more complex, students master “the distributive property”, “collecting like terms”, and perhaps even procedures made more memorable with acronyms like “FOIL”.

When learning to multiply binomials and polynomials, people often focus more on the process than the reasoning behind it – which can makes things feel complex. Once you understand the reasoning, multiplying polynomials will hopefully become straightforward. And with a small measure of melodrama, I will describe “FFFT!”, a trivial technique with a silly name that can help make multiplying polynomials easy.

Way back in elementary school, perhaps in first grade, you were probably taught to multiply two integers:

8\cdot 7\\*~\\*=56

From there, you were probably asked to learn your multiplication tables. It is extremely useful, particularly when studying algebra, to know your multiplication tables by rote. This “reflex knowledge” will help you work faster, rely on a calculator less, and verify that solutions are correct with greater speed and confidence. No matter how old you may be, no matter whether you are still in school or not, I recommend mastering any gaps your multiplication tables.

Having said that, I confess that Continue reading Multiplying Polynomials and FFFT!

Angle Measures

Suppose nobody had ever thought of measuring the size of an angle, and someone asked you “How can I describe the size of an angle?” What approach might you take in answering this question?

You might start by arbitrarily picking some angle, any angle, such as angle ABC in the image below, and call its measure “1”. All other angles could be
Continue reading Angle Measures

Algebra Intro 1: Numbers and Variables

This post begins a series intended to help introduce or re-introduce some of the core concepts of Algebra. It is often very helpful to re-visit these concepts with students who may have memorized their way through previous math courses without slowing down to contemplate some of the concepts behind Algebra.

Numbers

Numbers are used in the English language as both nouns and adjectives. However, the only physical instance of a “number” in our world is a symbol or group of symbols such as “23”. Most of the time, we seem to use numbers as adjectives: “Look at the two trees.” Continue reading Algebra Intro 1: Numbers and Variables

Algebra Intro 2: Addition

Mathematical thinking probably started with addition. Someone may have combined two piles of bricks, and wondered how many were in the single large pile. Addition is the mathematical term that describes “joining quantities together”.

Properties of Addition

Looking at the patterns that can occur when quantities are joined together, you might have noticed that it does not matter whether 2 bricks are added to a pile of 3, or 3 bricks are added to a pile of 2… either way, we still end up with a pile of 5. So, the order in which we add two quantities does not change the result. This probably makes intuitive sense to you when you visualize the situation above and the final pile that results. Continue reading Algebra Intro 2: Addition

Algebra Intro 3: Subtraction

Once addition has been explored a bit, it leads pretty naturally to a new question: if there are three bricks in a pile, how many bricks do I need to add to it so that there will be five in the pile?

Our addition problems were all phrased using a pattern like

number + number = what?

and the question above rearranges it a bit:

number + what? = number
or
what? + number = number

This question is usually asked as “What is the difference between Continue reading Algebra Intro 3: Subtraction

Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites

Negative Numbers

Negative numbers are something new and interesting to think about. What do they mean?

They arose from changing the order in which we subtracted two numbers. While we usually think of a “difference” by starting our thought process with the larger number, when we fail to do that and try to Continue reading Algebra Intro 4: Negative Numbers, Zero, Absolute Values, and Opposites

Algebra Intro 5: Addition, Subtraction, and Terms

What happens when problems involve both addition and subtraction? Addition is both associative and commutative, and subtraction is neither…

One solution is to follow the order of operations (parentheses, exponents, multiplication, division, addition, subtraction) working from left to right in the event of a “tie”. This will always produce the intended result.

However, the order of operations does not provide any guidance about how Continue reading Algebra Intro 5: Addition, Subtraction, and Terms

Algebra Intro 6: Multiplication

We have all known our multiplication tables for years, and have successfully answered questions like “what is 6 times 7?”, but do we really understand what multiplication represents?

Repeated Addition

One interpretation of multiplication, which only works when multiplying by an integer, is “repeated addition”. From this perspective, “6 times 7” is a compact way to Continue reading Algebra Intro 6: Multiplication

Algebra Intro 7: Properties of Multiplication

Properties Of Multiplication

Do the patterns that applied to addition also apply to multiplication… do the following all produce the same result?

3 \cdot 5 \cdot 7 \\*3 \cdot (5 \cdot 7) \\*(3 \cdot 5) \cdot 7

After carefully following the order of operations, we see that they all result in a value of 105. Therefore, it is reasonable to conclude that multiplication is associative, similar to addition.

Continue reading Algebra Intro 7: Properties of Multiplication

Algebra Intro 8: Division

The last “arithmetic” operation introduced in school is usually division. While multiplication allows us to calculate the total needed for a group when a fixed quantity is required for each person, division allows us to determine how much each person will  get when a fixed quantity is divided equally among all in a group.   Division is the inverse of multiplication.

But how does one go about dividing?  Should we use our knowledge of Continue reading Algebra Intro 8: Division