| Rule Name | Formula | Examples |
| 1. Multiplication Law (Same Base) | $a^m \times a^n = a^{m+n}$ | $2^5 \times 2^{10} = 2^{15}$ |
| 2. Division Law (Same Base) | $\frac{a^m}{a^n} = a^{m-n}$ | $\frac{5^7}{5^2} = 5^5$ |
| 3. Power of a Power | $(a^m)^n = a^{m \times n}$ | $(7^5)^2 = 7^{10}$ |
| 4. Power of a Product | $(ab)^n = a^n b^n$ | $(2x)^3 = 2^3 \cdot x^3 = 8x^3$ |
| 5. Power of a Quotient | $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ | $\left(\frac{2}{3}\right)^2 = \frac{4}{9}$ |
| Rule Name | Formula | Key Notes & Examples |
| 6. Zero Exponent Rule | $a^0 = 1$ (where $a \neq 0$) | $100^0 = 1$. Note: $0^0$ is undefined. |
| 7. Negative Exponent Rule | $a^{-n} = \frac{1}{a^n}$ | $3^{-4} = \frac{1}{3^4} = \frac{1}{81}$. |
| 8. Reciprocal Rule | $\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^n$ | $\left(\frac{3}{4}\right)^{-3} = \left(\frac{4}{3}\right)^3 = \frac{64}{27}$. |
If the bases are equal, the exponents must be equal.
Rule: If $a^m = a^n$, then $m=n$ (where $a \neq -1, 0, 1$).
Example: If $2^{x+7} = 128$, find $x$.
Since $128 = 2^7$, we set $x+7 = 7$.
$x = 0$.
The sign of a negative base depends on whether the exponent is even or odd.
$(-1)^{\text{even number}} = 1$.
$(-1)^{\text{odd number}} = -1$.
$(-a)^n = a^n$ when $n$ is even.
$(-a)^n = -a^n$ when $n$ is odd.
Numbers can be expressed as a product of powers of prime factors.
Example: Express $432$ as a product of powers of prime factors.
$432 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 3$
$432 = 2^4 \times 3^3$
To compare $2.7 \times 10^{12}$ and $1.5 \times 10^8$, align the exponents.
$2.7 \times 10^{12} = 27000 \times 10^8$.
Since $27000 > 1.5$, we find $2.7 \times 10^{12} > 1.5 \times 10^8$.