Programmer's Reference

Real numbers include:

1. Rational Numbers
2. Irrational Numbers

Rational Numbers include:

1. Integers
2. Whole Numbers
3. Natural Numbers (a.k.a. counting numbers}

Integers include:

1. Whole Numbers {0, 1, 2, 3, …}
2. Natural Numbers (a.k.a. Counting Numbers) {1, 2, 3, …}

Whole Numbers include:

1. Natural Numbers (a.k.a. Counting Numbers) {1, 2, 3, …}

Irrational Numbers include squareroots (√) and pi (ϖ).

This article discusses the practical application of exponent rules and how they relate to business and finance scenarios.

A common use of exponents in business and personal finance is the investment of money at a fixed rate for a variable period of time.  For example, let’s say we have $10,000 we want to invest.  A CD yields a 6% annual rate, so we decide we want to invest it for 20 years since.  What will be it’s value when the CD matures?

You may already know that the formula to compute this simple interest calculation is A = P(1 + i)ⁿ . Where P = $10,000, i = the 6% interest rate and n is the 5 years you can invest your money for in the CD.

However, it is useful to understand the mechanics of the formula because many variables can come into play later.  For example, if you felt the yield was good and were given the option to roll over the CD 3 times, how would you write that as an equation and solve it?  Below are the common rules for exponents.

Below is the list of the rules for exponents:

1. The Product Rule for exponents
2. The Zero Exponent Rule
3. The Quotient Rule for Exponents
4. The Power of a Power Rule for Exponents
5. The Power of a Product Rule for Exponents
6. The Power of a Quotient Rule for Exponents

The Product Rule for Exponents:

When a is a real number and m and n are positive integers then the following applies:

a^m * aⁿ = a^m+n. 

1. 5⁴ * 5³ = 5⁸
2. r² * r⁴ = r⁶
3. 2y³ * 5y⁹ = 10y¹²

 Note that neither bases or exponents are multiplied.  Exponents are added and similar bases remain as one.  This may seem a little odd when comparing example 1 and 3, but think about it this way, in example one, 5 is our base.  Just like we did not change r, in example 2, we do not change 5 from being 5.  However, in example 3, we can consider y as our base and multiply 2 and 5 to note that we now have 10 letter y to the 12th power (12 came from the adding the exponents 3 and 9).

The Zero exponent rule

Simply remember that any number to the zero power is 1.

Examples:

1. 5⁰ = 1
2. (12xy)⁰ = 1
3. 4⁰ + 7⁰ = 2

Example two illustrates the order of operations.  We do what is in parenthesis first, followed by exponents.  Therefore, any calculation could have taken place inside the parenthesis and resulted in any number, however when the exponent of 0 is introduced, the answer becomes 1.

The quotient rule for exponents

Similar to subtraction, the exponent in the denominator is subtracted from the exponent in the numerator.  This results in the base to the power of the difference in exponents.  When coefficients are present, you can reduce and eliminate them in the quotient as necessary.  Sometimes when doing this, you may be left with a fraction equation to work with in a simple form.

Example:

1. x⁷ / x⁴ = x³
2. 2x¹⁰ / 4x³ = (1/2)x³

The answer in the second example illustrates the fraction.  Since this article is being published on the web, the one-half was noted as 1 over two and enclosed in parenthesis.

The power of a power rule

WIth this rule, when an exponent is raised to another exponent, then both exponents are multiplied to become the new exponent.  The same treatment of coefficients applies to this rule as it did to the last one.

Examples:

1. (2³)³ = 2⁹ = 512
2. (x³)⁴ = x¹²
3. 3x⁸(x³)⁶ = 3x²⁶
4. -10(r⁴)³ / 5r² = -2r¹⁰

The power of a product rule

With n as a positive integer and a and b real numbers, then (ab)ⁿ – aⁿ * bⁿ

Example:

2x⁴ = 2⁴ * x⁴ = 16x⁴

Here, each variable and coefficient will break apart from each other, take the value of the exponent and simplify itself the best way possible before returning to you as a simplified equation.

The power of a quotient rule

(y/3)³ = (y³/3³) = y³/27

The quotient rule applies when nonzero real numbers are used as the variables and the nonnegative integer is the exponent.

That concludes todays reference of the rules of expressions.  In the future we will revisit these methods under some real life work problems.

The method you use to draw your graph depends on the form of your equation.

slope intercept form:  y = mx + b

standard form:  Ax + By = C

For slope intercept form (y = mx + b) set your x = 0 and by default your y will be equal to b.  This will be your starting point.  From there you can use your slope to plot the next point.  Slope is equal to rise/run or y/x, so you would go up y places on the graph and across x places.  Draw a straight line between your two points.

For standard form (Ax + By = C), to get two coordinates on your graph set your x equal to zero to get your first coordinate.  Your first coordinate will be equal to (0,y).  Next set your y = 0 to get your next coordinate.  The second coordinate will be (x, 0). Plot the points and connect the line.