A power like 2⁵ just means “2 multiplied by itself 5 times”. Because of that, multiplying and dividing powers becomes simple arithmetic on the little numbers up top (the exponents).
• (aᵐ)ⁿ = aᵐⁿ (power of a power → multiply the powers)
• a⁰ = 1 and a⁻ⁿ = an1
The master move for equations: rewrite every number over the same base. Once the bases match, the powers must be equal. Example: 9ˣ⁺¹ = 27ˣ⁻¹ → (3²)ˣ⁺¹ = (3³)ˣ⁻¹ → 2(x+1) = 3(x−1) → x = 5.
For a sum of powers you cannot merge exponents — instead factor out the smallest power.
⚡Laws of IndicesPick a base & two powers
base a
power m
power n
aᵐ × aⁿ = aᵐ⁺ⁿ
2⁵ × 2³ = 2⁸ = 256
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
2⁵ ÷ 2³ = 2² = 4
(aᵐ)ⁿ = aᵐⁿ
(2⁵)³ = 2¹⁵ = 32768
2⁵ × 2³ = 2 to the power (add the powers: 5 + 3).
📝Practice Questions
Q1Simplify: 2⁵ × 2³ ÷ 2⁶
Q2If 3ˣ = 81, then x is:
Q3The value of (2⁰ + 3⁰ + 4⁰) is:
📚
Real exam questions — Powers & Exponents
2 question types · 6 solved examples from real SSC papers
Bring everything to one base and the laws of indices do the rest — add, subtract, or factor the exponents.
How to solve this type
Use the basic laws: aᵐ × aⁿ = aᵐ⁺ⁿ (multiply → add the powers), aᵐ ÷ aⁿ = aᵐ⁻ⁿ (divide → subtract the powers), (aᵐ)ⁿ = aᵐⁿ (power of a power → multiply the powers), a⁰ = 1, and a⁻ⁿ = 1/aⁿ. To compare two sides or solve an equation, first rewrite every number over the SAME base (or the same power); once the bases match, the exponents must be equal. If you are only asked to simplify, just add and subtract the exponents of the common base.
