Arithmetic/Introduction to Exponents

What Are Exponents

Exponents are a short hand that tells us how many times we should multiply a number with itself. If we have to write out every multiplication expressions could get quite long. For example, the number 5×5×5×5×5×5 is often shortened to 56. The number 6 is called an exponent. It tells us how many times we should multiply 5 by itself.

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 ${\displaystyle 5*2=5+5}$ , ${\displaystyle 5^{2}=5*5}$ . Overall, ${\displaystyle x^{2}=x*x}$ . 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, ${\displaystyle 2^{3}=2*2*2=8}$ , while ${\displaystyle (-2)^{3}=-2*-2*-2=-8}$ .

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. Say we have some powers of ${\displaystyle x}$ , for example, ${\displaystyle x^{a}}$  and ${\displaystyle x^{b}}$ . If we want to figure out the product of these, ${\displaystyle x^{a}*x^{b}}$ , we can see that it is actually ${\displaystyle x}$  multiplied by itself ${\displaystyle a}$  times, multiplied by itself ${\displaystyle b}$  more 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, ${\displaystyle x^{a}*x^{b}=x^{a+b}}$ .

In the same way, ${\displaystyle \left({\frac {x^{a}}{x^{b}}}\right)}$  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 ${\displaystyle \left({\frac {x^{a}}{x^{b}}}\right)}$  will turn to ${\displaystyle \left({\frac {x^{a-b}}{1}}\right)=x^{a-b}}$
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