Every laws of exponents calculator — the one on this site included — is exactly eight rules deep. There is no ninth rule hiding underneath, and no calculation a law of exponents calculator performs that cannot be traced back through these eight to a plain chain of multiplications. This guide states each law once, shows the chain that forces it to be true, works one example all the way to a number, and names the standard wrong answer — because the wrong answers are as regular as the rules, and recognising them is most of what “being good at exponents” means. For quick answers, the Exponent Calculator applies all eight instantly; this page is for making them stick.

Why there are exactly eight

An exponent counts multiplications: 5³ is 5 × 5 × 5. Every law below is what happens when that counting meets one ordinary operation — multiplying two powers, dividing them, raising a power to another power, and so on. There are eight combinations worth naming, and each one, unfolded, is just the chain written out longhand and re-counted. That is also the audit trail: whenever an exponent rules calculator gives you an answer you doubt, you can unfold the chain by hand for the small case and check which law was used. The laws never contradict the chain; they are the chain, compressed.

1. The product rule: add the exponents

Rule: b^m × b^n = b^(m + n). Worked: 5² × 5³. As chains, that is (5 × 5) × (5 × 5 × 5) — five factors of 5 standing in a row — so the answer is 5⁵. Check the numbers: 5² = 25, 5³ = 125, and 25 × 125 = 3,125, which is indeed 5⁵. The standard wrong answer multiplies the exponents instead, producing 5⁶ = 15,625 — five times too large. Exponents are added because factors are being joined, not composed.

2. The quotient rule: subtract the exponents

Rule: b^m ÷ b^n = b^(m − n). Worked: 10⁶ ÷ 10². Division cancels factors: a chain of six 10s divided by a chain of two 10s loses two factors, leaving four — 10⁴. Check: 1,000,000 ÷ 100 = 10,000 = 10⁴. The standard wrong answer divides the exponents (6 ÷ 2 = 3) and reports 10³ = 1,000 — the rule for a different operation entirely (see law 3’s cousin, roots). Cancellation subtracts; nothing about division of powers ever divides the counters.

3. Power of a power: multiply the exponents

Rule: (b^m)^n = b^(m × n). Worked: (3²)³. The outer exponent repeats the whole inner power: (3²) × (3²) × (3²), and each 3² is two factors of 3, so the chain holds six — 3⁶. Check: 3² = 9, and 9³ = 729, matching 3⁶ = 729. This is the one law where the exponents genuinely multiply, and the standard wrong answer imports law 1 and adds instead: 3^(2 + 3) = 3⁵ = 243. If you remember only one pairing, remember this: joining chains adds; repeating chains multiplies.

4. Power of a product: distribute over multiplication

Rule: (a × b)^n = a^n × b^n. Worked: (2 × 5)³. The product 2 × 5 is made three times: (2 × 5) × (2 × 5) × (2 × 5), and regrouping gathers three 2s and three 5s — 2³ × 5³. Check: 10³ = 1,000, and 2³ × 5³ = 8 × 125 = 1,000. The standard wrong answer distributes over only part of the product — 2 × 5³ = 250 — applying the power to one factor and forgetting the other. The exponent outside brackets belongs to everything inside them.

5. Power of a quotient: distribute over division

Rule: (a ÷ b)^n = a^n ÷ b^n. Worked: (6 ÷ 3)². The quotient is 2, and 2² = 4. Distributing first gives the same: 6² ÷ 3² = 36 ÷ 9 = 4. The standard wrong answer subtracts the separate powers — 6² − 3² = 27 — confusing “a power of a quotient” with “a difference of powers”, two expressions that share no machinery at all.

6. The zero exponent: everything nonzero becomes 1

Rule: b⁰ = 1, for any nonzero b. Worked: 12⁰ = 1. This is not a convention pasted on; law 2 forces it. 12³ ÷ 12³ must equal 1 (anything divided by itself), and the quotient rule writes it as 12^(3 − 3) = 12⁰ — so 12⁰ and 1 are the same number, or the quotient rule breaks. The standard wrong answer is 0, usually by confusion with “12 × 0”. Multiplying by zero and raising to the zero power are unrelated operations that happen to share a digit. (The base itself may not be zero: 0⁰ stays undefined, because the two rules that meet there disagree.)

7. The negative exponent: flip to a reciprocal

Rule: b^(−n) = 1 ÷ b^n. Worked: 3⁻⁴. Law 2 again: continuing 3², 3¹, 3⁰ downward, each step divides by 3, so 3⁻⁴ = 1 ÷ 3⁴ = 1 ÷ 81 = 0.012346. The standard wrong answer attaches the minus sign to the result and writes −81. The sign in the exponent is an instruction — “reciprocate” — not a sign for the answer; 3⁻⁴ is a small positive fraction, and every negative exponent on a positive base is.

8. The fractional exponent: a root in disguise

Rule: b^(1/n) = the nth root of b, and b^(m/n) is that root raised to m. Worked: 64^(5/6). The denominator names the root: the sixth root of 64 is 2, since 2⁶ = 64. The numerator then applies an ordinary power: 2⁵ = 32. So 64^(5/6) = 32. Law 3 is what forces this reading — (64^(1/6))⁶ must return 64, and only the sixth root does that. The standard wrong answer multiplies the base by the fraction (64 × 5/6 ≈ 53.3), treating the exponent as a scaling factor. An exponent is never a multiplier of the base; it is a count — here, a count of roots and powers combined.

The laws under stress: negative bases, fractional bases, two laws at once

The eight laws were stated for friendly numbers. They are worth more than that: they hold — unchanged — for the unfriendly ones, and checking three hard cases is the fastest way to be sure you understand them rather than merely recognise them.

Negative bases. (−2)³ × (−2)² is a product of powers of the same base — the base happens to be −2 — so law 1 adds the exponents: (−2)⁵ = -32. Worked the long way, (−2)³ = −8 and (−2)² = 4, and −8 × 4 = -32: the law and the chain agree, and the sign takes care of itself through the parity of the total count — five negative factors, an odd count, a negative result. No special negative-base law is needed; there is only the one law, applied to a negative base.

Fractional bases. (1/2)² × (1/2)³ adds exponents in exactly the same way: (1/2)⁵ = 1/32 ≈ 0.03125. Longhand: 1/4 × 1/8 = 1/32. The laws were never about whole numbers; they are about counting factors, and a factor of 1/2 counts exactly like a factor of 5. This is also why the negative-exponent law (law 7) is not a separate universe: 2⁻⁵ and (1/2)⁵ are the same number, 0.03125, wearing the two notations — the flip in law 7 is precisely the move that swaps one for the other.

Two laws in one expression. (2³ × 2²)² invites both law 1 and law 3, and the expression is a small test of whether “the brackets decide the order” was understood. Route one — product first, then power of a power: inside the brackets, 2³ × 2² = 2⁵; squared, (2⁵)² = 2¹⁰ = 1,024. Route two — distribute the outer power first, then use the product rule: (2³)² × (2²)² = 2⁶ × 2⁴ = 64 × 16 = 1,024. Two different decompositions, one answer, because the laws are consistent views of a single chain of ten 2s: 8 × 4 = 32 inside, then 32² = 1,024 — and indeed 2¹⁰ = 1,024. When two routes through the laws ever disagree, the error is in the applying, not the laws.

The eight, on one line each

All eight laws with the worked checks from the sections above. Any exponent calculation, however large, decomposes into these moves.

One habit ties the eight together: when two laws could apply, the brackets decide the order. (3²)³ is law 3 because the outer power acts on the whole inner power; 3² × 3³ is law 1 because two separate powers are being multiplied. Read the expression’s structure first — what is inside brackets, what stands alone — and the correct law selects itself. And when a result from any calculator surprises you, audit it exactly the way this page audited each law: pick the claim apart into a checkable multiplication. A result like 7⁴ × 7² = 7⁶ can be verified without trusting law 1 at all — 7⁴ = 2,401, 7² = 49, and 2,401 × 49 = 117,649, which is 7⁶ = 117,649 by direct count. The chain is the court of final appeal; the laws, and the calculator built on them, are only ever its shortcuts.