Laws of Integral Indices

We use index notation in repeated multiplication of numbers and unknowns. For any number $a$ and any positive integer $n$, we write:

Laws of Positive Integral Indices

If $m$ and $n$ are positive integers, then we have:

• $a^m \times a^n = a^{m+n}$
• $\frac{a^m}{a^n} = a^{m-n}$ where $m > n$
• $\frac{a^m}{a^n} = \frac{1}{a^{n-m}}$ where $m < n$
• $(a^m)^n = a^{mn}$
• $(ab)^n = a^n b^n$
• $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ (where $b \neq 0$)

Zero Exponent

For any non-zero number $a$, $a^0 = 1$.

Negative Exponents

For any non-zero number $a$ and positive integer $n$, $a^{-n} = \frac{1}{a^n}$.

Applying the Laws

The laws of indices also apply to negative and zero indices.

Scientific notation is a way to express very large or very small numbers. A number is written in scientific notation if it is in the form:

where $1 \le a < 10$ and $n$ is an integer.

Examples

• 20,000 = $2 \times 10^4$
• 3,720,000 = $3.72 \times 10^6$
• -6,300 = $-6.3 \times 10^3$
• 0.0003 = $3 \times 10^{-4}$
• 0.00000012 = $1.2 \times 10^{-7}$

Denary System (Base-10)

In the denary system, we use 10 numerals: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The place values are powers of 10.

Binary System (Base-2)

In computers, the binary system is used. It consists of only 2 numerals: 0 and 1. The place values are powers of 2.