Arithmetic/Introduction to Exponents

What Are Exponents

Exponents are to multiplication what multiplication is to adding. Just like how multiplying is like adding several times, i.e. 4*5 is the same thing as 4+4+4+4+4 (4 added together five times). In the same way, exponents function like this, except with multiplication, i.e. 4^5 would be 4*4*4*4*4 (four multiplied together five times). While multiplying two numbers is written as $ab$ or $a*b$, exponentiating them is written as $a^b$.

A Focus on Squaring Numbers

In multiplication, there are some common names when we use small numbers. For example, multiplying by two is called "doubling", and by three is "tripling". In the same way, exponentiating with two is called "squaring" and by three is called "cubing".

So, just how $5*2=5+5$, $5^2=5*5$. Overall, $x^2=x*x$. So, 6^2 is 6*6 (36), and 10^2 is 10*10 (100).

You might also notice that the square of a number, as we have learned it, is always positive. For example, 2 squared is 2*2 which is 4; at the same time, -2 squared is -2*-2, which is also 4. Actually, -x^2 and x^2 equal the same thing.

A Focus on Cubing Numbers

Unlike squaring numbers, cubing them (raising them by three) preserves the original numbers' sign. For example, $2^3=2*2*2=8$, while $(-2{)}^3=-2*-2*-2=-8$.

In fact, you might notice that when you raise a number to any even power (2, 4, 6, ...), the sign must be positive, whereas when you raise it to an odd degree (1, 3, 5, 7, ...) the sign will be the same as the original number's.

Properties of Exponents

There are, however, several useful tricks for manipulating exponents. For example, say we have $x^a$ and $x^b$ are some numbers. If we want to figure out the product of these, $x^a*x^b$, we can see that it is actually x times itself a times, times itself b times.

In other words, there are a x's in the first term, and b x's in the second, and multiplying the two means that there will be a+b x's in the answer. Therefore, $x^a*x^b=x^{(}a+b)$.

In the same way, $\left(\frac{x^a}{x^b}\right)$ means that there are a x's on the top, divided by b x's on the bottom. As you may see, some of the x's on the bottom will cancel those on the top, particularly, b. This means that the fraction $\left(\frac{x^a}{x^b}\right)$ will turn to $\left(\frac{x^{a-b}}{1}\right)=x^{a-b}$