1.7+Exponents

Section 1. 7 - Exponents

 * Exponents are Repeated Multiplication.**

Exponents are small numbers (usually ), in superscript just after a number or a variable. The raised number is the exponent, the number or variable is called the base.



Some other vocabulary terms:


 * Power **- another name for exponent. You will often see it described as "raised to the third power" or "raised to a power of 5".


 * Squared **- something raised to an exponent of 2. "Square units" often refers to the area of a shape, where you are multiplying a unit of measurement, such as feet, by feet again. 6 feet x 8 feet = 6 x 8 x feet x feet = 48 feet squared.


 * Cubed **- raised to the power of 3.

The exponent has a special meaning - it means multiply the base, by itself, as many times as the exponent. In the example above, it means 5 x 5 x 5 = 125.




 * Exponents Only Affect the Thing Right in Front of It! **

By thing I mean number or variable. Unless it is in parentheses, the exponent only affects the number or variable right in front of it. Not a negative sign, not other numbers or exponents that might be in the same term. Look carefully at the examples below and understand this concept.




 * Powers of 10**

Think about what happens when the base is ten.



The pattern is very easy - as long as the base is ten, just put a 1 and as many zeroes as the exponent. This will be amazingly useful in a later section on Scientific Notation.

If you notice, 10 to the first power is the same as 10. Normally we don't write the exponent if it is 1, because it doesn't really mean anything different, but it still has an exponent or power.

By the way, if you follow the pattern backwards, wouldn't ten to the zero exponent be a 1 with no zeroes? So then ten to the zero would be equal to 1. It is, but we'll talk about that later.


 * Perfect Squares**

Perfect squares are formed when any whole number, except zero, is squared. 1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9. 1, 4, 9, 16 . . . are all perfect squares because they are created from whole numbers. It is possible to have other squares. For example, if you multiplied 1.5 x 1.5, the answer is not 3 -- it's 2.25. 2.25 is not a perfect square because it was formed with a decimal number, or a fraction, and not a whole number.


 * Square Roots and the Radical**

Square Roots are the opposite of squares. The sign for a square root is called a radical, and it looks something like the division sign used to do long division.

The number written beneath the radical is the square. Essentially, the sign means, "What number, multiplied by itself (another way to say squared), equals this number?"



Times tables can help with most square roots that are perfect squares. If the number is too large, such as 625, you might have to do some quick multiplication problems to find that 25 x 25 = 625.


 * Square Roots without a Perfect Square**

The problem is worse, however, when you are trying to find the square root of numbers that are not perfect squares. That is why they have a radical button on most calculators, because otherwise it requires a lot of trial and error multiplication to even get close. For example, what is the square root of 50? What number, times itself, equals 50? If you didn't have a calculator, or couldn't use one, you would have to do trial and error by hand.



1.1 Place Value 1.2 Rounding and Estimating Numbers 1.3 Addition 1.4 Subtraction 1.5 Multiplication 1.6 Division 1.7 Exponents 1.8 Order of Operations 1.9 Prime Numbers
 * Where to from here?**
 * Chapter One Review**
 * Chapter One Homework**