Arithmetic/Exponents

Exponents

Exponents or 'powers' are a process of repeated multiplication, in much the same way that multiplication is a process of repeated addition.

Exponents are normally written in the form $a^{b}$, where $a$ is the base and $b$ is the exponent. In contexts where superscripts are not available, such as in many contexts in computers, $a^{b}$ is commonly written as "a^b" or less often as "a**b". If you're not familiar with algebra, you can just imagine the letters a and b as representing numbers. We pronounce $a^{b}$ as a to the power of b, a to the b or a exponent b.

Integer exponents

When the exponent is a positive integer, then it is just a simple case of multiplying the base by itself a certain number of times. For example,

$3^{4}=3\times 3\times 3\times 3=81\,$

Here, 3 is the base, 4 is the exponent (written as a superscript), and 81 is 3 raised to the 4th power. Notice that the base 3 appears 4 times in the repeated multiplication, because the exponent is 4.

Some more examples:

${\begin{matrix}12^{2}=12\times 12=144\\2^{8}=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2=256\\1^{5}=1\times 1\times 1\times 1\times 1=1\end{matrix}}$

Multiplying exponents

If you have two or more exponents with the same base, then multiplying them has the same effect as adding their exponents.

For instance $(a^{b})*(a^{c})\,$ is the same as $a^{b+c}\,$. For example,

$(3^{4})*(3^{2})=(3*3*3*3)*(3*3)=(3*3*3*3*3*3)=(3^{6})=(3^{4+2})\,$

Dividing exponents

If you have two or more exponents with the same base, then dividing them has the same effect as subtracting their exponents.

For instance $(a^{b})/(a^{c})\,$ is the same as $a^{b-c}\,$. For example,

$(3^{6})/(3^{2})=(3*3*3*3*3*3)/(3*3)=(3*3*3*3)=(3^{4})=(3^{6-2})\,$

Exercises

What is?

1

$4^{3}=$

2

$3^{4}=$

3

$1^{250}=$

4

${250}^{1}=$

Write these numbers as powers of 2

5

$128=2^{\wedge }$

6

$8=2^{\wedge }$

7

$1024=2^{\wedge }$

8 What is?

$(2^{3})*(2^{2})=$

9 What is?

$(2^{6})/(2^{2})=$

10 Harder:

Why does $3^{0}=1\,$ (clue: think about ${3^{2}}/{3^{2}}\,$, for example) (answer on paper)

Negative exponents

Negative exponents work slightly differently. Let's say you want to calculate $3^{-2}$. To do that, you take $1/3^{2}$ to get your answer. We do the exponent first, see Order of Operations

$3^{-2}={\frac {1}{3^{2}}}={\frac {1}{9}}$

The commutative property doesn't apply in exponents. See for yourself! Try to calculate 2³, and then see if it is the same as 3² (answer here). The distributive and associative properties don't apply either.

Exponents do, however, have their own set of axioms that they consistently follow. Consistent with the preceding examples, one can state generally that:

$(a^{b})\times (a^{c})=a^{b+c}\,\,\,\,$ and

$(a^{b})/(a^{c})=a^{b-c}\,\,\,\,$

It's also easy to see that ${\begin{matrix}(a^{b})^{c}=a^{b\times c}\end{matrix}}$

Fractional exponents

So far, we have only seen exponents as whole numbers, but exponents can be fractional as well. With a fractional exponent, the numerator acts as a normal whole-number exponent, while the denominator acts as a root.

In general, $a^{p/q}={\sqrt[{q}]{a^{p}}}\,\,\,\,$ for any real number $q$ ≠ 0.

Let's look at $8^{2/3}$ as an example. First, we raise 8 to the power of the numerator, 2. Then, since the denominator is 3, we take the third root of this number. The expression is read as the cubed root of eight squared, and written as:

$8^{2/3}={\sqrt[{3}]{8^{2}}}={\sqrt[{3}]{64}}=4$

It should then be evident that when the numerator of the fractional exponent is 1, the expression is a simple root. That is, $1/2$ is a square root, $1/3$ is a cubed root, $1/4$ is a fourth root, etc.

For example, $9^{1/2}$ would be read as the square root of nine, and written as:

$9^{1/2}={\sqrt[{2}]{9^{1}}}={\sqrt {9}}=3$