What are Exponents?

Exponents are a way of writing repeated multiplication. For example, let's consider this expression:

$$3+3+3+3+3+3$$

Not very pretty, is it? You (should have) learned that this can also be written as

$$3\cdot6$$

which is 3 added to itself 6 times. Similarly, if we have an expression like this:

$$3\cdot3\cdot3\cdot3\cdot3\cdot3$$

We can write it using an exponent as

$$3^6$$

(read as 3 to the power of 6, 3 to the 6th power, or 3 to the 6th).

In this example, the "3" is called the base, and it is the number being multiplied by itself. The "6" is called the exponent, and it is the number of times that the base is multiplied by itself. This notation is called exponential form. A base together with its exponent are called a power.

To type a power on a computer, type {base}^{exponent}. For example, for $3^6$ we would type 3^6.

Your turn! Write the following in exponential form:

$\displaystyle{2\cdot2\cdot2\cdot2}=$

Hint

What is exponential form?

Hint

Which number is the base? How about the exponent?

Show/Hide Solution

2 is the base, and 4 is the exponent. In exponential form, this is $2^4$.

Tips

1. For multiplication, we can think of it as a number being added multiple times to 0. $$x\cdot 4=0+x+x+x+x$$ However, for exponents, we have to think of it as being multiplied to 1 instead of 0. $$x^4=1\cdot x\cdot x\cdot x\cdot x \\ \neq 0\cdot x\cdot x\cdot x\cdot x$$ This is because any number added to 0 is the same number, and any number multiplied by 1 is the same. These special numbers are called identity elements (additive identity and multiplicative identity).


2. In general, $$x^y \neq y^x$$ For example, $$2^3=8\neq3^2=9$$ Can you find an example where $x^y=y^x$?
   Solution

   Try $2^4$ and $4^2$.


3. Remember your order of operations when working with exponents. Exponents come before all other operations except parentheses! $$x\cdot y^z = x\cdot (y^z)\neq (x\cdot y)^z$$


4. If an exponent is a power, evaluate the expression from top-right to bottom-left. $$x^{y^z} = x^{(y^z)}\neq (x^y)^z$$