1. The Product Rule

• When multiplying two powers with the same base, keep the base and add the exponents.
• $a^m \times a^n = a^{m+n}$

Example : $2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$

2. The Quotient Rule

• When dividing two powers with the same base, keep the base and subtract the exponents.
• $a^m \div a^n = a^{m-n}$ , where $a \neq 0$

Example : $7^6 \div 7^2 = 7^{6-2} = 7^4 = 2401$

3. The Power of a Power Rule

• When a power is raised to another power, keep the base and multiply the exponents.
• $(a^m)^n = a^{m \times n}$

Example : $(2^3)^4 = 2^{3 \times 4} = 2^{12} = 4096$

4. The Power of a Product Rule

• To raise a product to a power, raise each factor to that power.
• $(ab)^n = a^n b^n$

Example : $(2x)^3 = 2^3 x^3 = 8x^3$

5. Zero Exponent

• Any nonzero base raised to the power 0 equals 1.
• $a^0 = 1$ , where $a \neq 0$

Example : $5^0 = 1$ ,   $(7x^2)^0 = 1$ (as long as $x \neq 0$)

6. Negative Exponent

• A negative exponent means "flip to the reciprocal and make the exponent positive".
• $a^{-n} = \dfrac{1}{a^n}$ , where $a \neq 0$

Example : $4^{-2} = \dfrac{1}{4^2} = \dfrac{1}{16}$

7. Combining Several Rules Together

• Complex expressions often need more than one rule — work step by step, simplifying the numerator and denominator first, then combining.

Example : $\dfrac{(3a^2b)^3 \times (2ab^2)^2}{(6a^2b^3)^2} = \dfrac{27a^6b^3 \times 4a^2b^4}{36a^4b^6} = \dfrac{108a^8b^7}{36a^4b^6} = 3a^4b$

8. WORKED EXAMPLES

Example 1 : Simplify $a^5 \times (a^3)^2 \div a^4$.

$(a^3)^2 = a^6$ (power rule) → $a^5 \times a^6 = a^{11}$ (product rule) → $a^{11} \div a^4 = a^7$ (quotient rule).

Example 2 : Find the value of $\dfrac{(3a^2b)^3 \times (2ab^2)^2}{6a^4b^3}$.

Numerator $= 27a^6b^3 \times 4a^2b^4 = 108a^8b^7$. Divide by $6a^4b^3$: $\dfrac{108a^8b^7}{6a^4b^3} = 18a^4b^4$.

Example 3 : A bacteria culture starts at 500 and doubles every day. Find the population after 5 days.

Population $= 500 \times 2^5 = 500 \times 32 = 16000$.