Exponents

Definition: The exponent of a number says how many times to use the number in a multiplication.

• Notation: $a^3$ means $a \times a \times a$.
• $a^n$ means multiply $a$ by $n$ times.
• $a$ is called the base.
• $n$ is called the exponent.

note

Negative exponent example:

$5^{-3}$ could be calculated like:

$$\frac{1}{5^3} = \frac{1}{5 \times 5 \times 5}=\frac{1}{125}$$

info

What if the Exponent is 0?

If the exponent is 0, then the answer is 1. (example $5^0 = 1$)

Laws of Exponents

• $(ab)^n = a^n \times b^n$
• $a^m \times a^n = a^{m + n}$
• $a^{m - n} = \frac{a^m}{a^n}$ (when $a \not= 0$)
• $a^{-n} = \frac{1}{a^n}$ (when $a \not= 0$)
• $a^0 = 1$
• $(a^m)^n = a^{m \times n}$

Important Exponents (powers of 2)

We often see a lot of powers of 2 in computer science. For example, Here are some prefixes:

Prefix	Multiply By	Number
kilo-	$2^{10}$	1,024
mega-	$(2^{10})^2$	1,048,576
giga-	$(2^{10})^3$	1,073,741,824
tera-	$(2^{10})^4$	1,099,511,627,776
peta-	$(2^{10})^5$	1,125,899,906,842,624
exa-	$(2^{10})^6$	1,152,921,504,606,847,000

Exponent Examples

• $2^{10}$ is the number of bytes in a kilobyte.
• $2^3 = 8$ is the number of bits in a byte.
• $2^{10} \times 2^{3} = 2^{10 + 3} = 8192$ is the number of bits in a kilobyte.

References

• https://www.mathsisfun.com/exponent.html