Can You Think Exponentially?

Changes, Changes, Changes  What types of new technologies exist today that didn’t exist when I was your age?  Video Video

In Other Words…  The world is changing exponentially!  As a result, we have to find a way to show very large numbers in a much easier way.  In other words, it would take quite a long time to write: 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 WOW!  Can’t there be a better way?

Yes – There is a Better Way!  Exponents are a shorthand way to show how many times a number, called the base, is multiplied times itself.  Exponents look like this: 2³  It would be: 2 x 2 x 2  The number 2 is the base. The 3 is the exponent.  A number with an exponent is said to be "raised to the power" of that exponent.

Here are some simple rules to use with exponents.  a1 = a Any number raised to the power of one equals the number itself.  For any number a, except 0, a0 = 1 Any number raised to the power of zero, except zero, equals one.  For any numbers a, b, and c, ab x ac = ab+c This multiplication rule tells us that we can simply add the exponents when multiplying two powers with the same base.

ALERT! These are mistakes that students often make when dealing with exponents. MMMMistake! Do not multiply the base and the exponent. 26 is not equal to 12, it's 64! MMMMistake! The multiplication rule only applies to expressions with the same base. Four squared times two cubed is not the same as 8 raised to the power two plus three. MMMMistake! The multiplication rule applies just to the product, not to the sum of two numbers.

Let’s Take a Look at Some Examples  Examples Examples  At home, if you want to try more, here is another place to practice. practice

Here’s a problem for you to solve: IIIIf a school can hold 1,200 students, how many days would it take to fill the school if you started with 2 students and increased exponentially by 3 each day? AAAAnswer: 2³ = 2x2x2 = 8 – Day 1 8888³ = 8x8x8 = 512 – Day 2 555512³ = 512x512x512 = 134,217,728 – Wow – That school Is already overcrowded by day 3! TTTThis demonstrates to students how things can increase so much faster exponentially.