Can you remember when you learned multiplication and discovered multiplication was a faster and shorter way to represent repeated addition? Well there is a faster and shorter way to represent repeated multiplication such as 2 x 2 x 2 x2. It is called exponential notation or exponential form. So, 2 x 2 x 2 x2 can be represented as 2 ^4. The “2” is called the base, and the “4” is referred to as the exponent. The base tells you what number is repeatedly multiplied, and the exponent tells you how many times to multiply.

Before we work any problems, let’s look at how to read the exponential form.
3 --- > three to the first power
3^1 --- > three to the first power
3^2 --- > three to the second power or commonly called three squared
3^3 --- > three to the third power or sometimes referred to as three cubed
3^4 --- > three to the fourth power
3^5 --- > three to the fifth power
And so on.

Exponent Tip: Any number raised to the zero power is always one. For example, 4^0 = 1; 123^0 = 1

Let’s try a few problems. Write the exponential notation for the following expressions.

Example: 4 x 4 x 4 = 4^3

Practice I
Write the exponential notation for the following expressions.
a) 9 x 9 x 9 =
b) 10 x 10 x 10 x 10 =
c) (-3) x (-3) x (-3) x (-3) x (-3) =

Practice II
Solve.
Example: 2^4 = 2 x 2 x 2 x 2 = 16

Note: (-5)^2 is different from -5^2 (-5)^2 = (-5)(-5) = 25 However - 5^2 requires you to view the “-“ as a signal to find the opposite of the value. In other words cover up or ignore the “-“ for a moment and simplify 5^2 then restore the negative sign. Another way to remember: if the “-“ is within the parentheses, raise everything enclosed. Otherwise, the negative sign is excluded from the “activity of being raised to a power.” Thus, -5^2 = -25.

Solve:
a) 3^4 =
b) 5^3 =
c) 11^2 =
d) 6^4 =
e) (-4)^3 =
f) (-7)^2 =
g) 5^0 =
h) 1^3 =
i) - 2^4 =





Answers:

Practice I:
a) 9^3
b) 10^4
c) (-3)^5


Practice II:
a) 81
b) 125
c) 121
d) 1296
e) -64
f) 49
g) 1
h) 1
i) -16

More Exponent Talk – Basic Exponent Rules
Exponents are especially helpful when dealing with variables. Any letter can represent a variable. However, x and y are commonly used. For instance, a better way to represent “yyyy” is y^4. The important thing to remember is to only combine exponents that have the same base and to follow some basic exponent rules.

1. Product Rule: Multiplication
Let’s look at (x^4)(x^5)
= xxxx + xxxxx
= x^9
Add the Exponents

2. Quotient Rule: Division
(x^6) / (x^2)
= xxxxxx – xx
= x^4
Subtract the Exponents

3. Power raised to a Power
(mn)^2 = m^2 n^2

(m^4)^3 = m^12
Multiply the Exponents