Chapter 4.2 — Exercise 4.1 — Laws of Exponents

The Six Laws of Exponents

• Product law: aᵐ × aⁿ = aᵐ⁺ⁿ — multiplying same-base powers adds their exponents.
• Quotient law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ — dividing same-base powers subtracts their exponents.
• Zero exponent: a⁰ = 1 — any nonzero base raised to the power 0 equals 1.
• Power of a product: aᵐ × bᵐ = (ab)ᵐ — same-exponent powers of different bases combine by multiplying the bases first.
• Power of a quotient: aᵐ ÷ bᵐ = (a/b)ᵐ — the same idea applied to division.
• Power of a power: (aᵐ)ⁿ = aᵐⁿ — raising a power to another power multiplies the exponents.

Alongside these: if aᵐ = aⁿ for the same base a, then the exponents themselves must be equal, m = n — a rule used constantly when a problem asks you to find an unknown exponent.

Question 1 — Negative Exponents

Each part applies a⁻ⁿ = 1/aⁿ directly:

• 4⁻³ = 1/4³ = 1/64.
• (-2)⁷ = -2 multiplied by itself seven times = -128 (an odd number of negative factors keeps the result negative).
• (3/4)⁻³ = (4/3)³ = 64/27 — flipping the fraction before cubing.
• (-3)⁻⁴ = 1/(-3)⁴ = 1/81 (an even power of a negative number is always positive).

Question 2 — Combining Powers of the Same Base

The product law handles all of these directly: (1/2)⁴ × (1/2)⁵ × (1/2)⁶ = (1/2)¹⁵, and (-2)⁷ × (-2)³ × (-2)⁴ = (-2)¹⁴. When the bases differ but the exponent is shared, the power-of-a-product law applies instead: 4⁴ × (5/4)⁴ = (4 × 5/4)⁴ = 5⁴ = 625.

Question 3 — Mixing Positive and Negative Exponents

(4⁻¹ × 3⁻¹) ÷ 6⁻¹ shows how the rules combine: 4⁻¹ × 3⁻¹ = 1/12, and dividing by 6⁻¹ is the same as multiplying by 6, giving 6/12 = 1/2. Keeping track of which exponent is negative and applying a⁻ⁿ = 1/aⁿ before combining anything else is what keeps a mixed expression like this manageable.

Question 4 — Multi-Step Simplifications

These questions chain several laws together in sequence. For 2⁻¹ + 3⁻¹ + 4⁻¹ × 3/4: first convert each negative power to a fraction (1/2 + 1/3 + 1/4 × 3/4), then combine using the LCM of the denominators before multiplying through — arriving at 13/16. The key habit here is applying one law at a time, fully simplifying after each step, rather than trying to combine everything in a single leap.

Question 6 — Finding an Unknown Exponent

These questions use the "equal powers mean equal exponents" rule in reverse. For (2/3)³ × (2/3)⁵ = (2/3)ⁿ⁻²: the product law simplifies the left side to (2/3)⁸, so matching exponents gives n − 2 = 8, and n = 10. For 7^(3n+1) × 49 = 7³: rewriting 49 as 7² first gives 7^(3n+1) × 7² = 7³, so 3n + 1 + 2 = 3, meaning 3n + 3 = 3 and n = 0.

Question 9 — Substituting Values

With m = 3 and n = 2: 9m² − 10n³ = 9(9) − 10(8) = 81 − 80 = 1. And mⁿ − nᵐ = 3² − 2³ = 9 − 8 = 1 as well — a reminder that mⁿ and nᵐ are generally different values, even though they happen to coincide here, and shouldn't be assumed equal without checking.

The Instinct This Exercise Builds

Every question above reduces to the same underlying question: are the bases the same, or is the exponent the same? Same base, different exponents being multiplied or divided → add or subtract exponents. Same exponent, different bases being multiplied or divided → combine the bases first. Recognising which situation you're in, before reaching for a specific law, is what makes the six rules feel like one connected idea rather than six separate ones to memorise.

Why the Product and Quotient Laws Are the Same Idea Twice

It helps to see the product and quotient laws as two faces of one underlying fact rather than two rules to memorise separately. aᵐ × aⁿ simply writes out a multiplied by itself m times, then n more times — (m+n) copies of a in total, hence aᵐ⁺ⁿ. Division does the reverse: aᵐ ÷ aⁿ cancels n of the copies of a in the numerator against the ones in the denominator, leaving (m−n) copies behind, hence aᵐ⁻ⁿ. Once you can picture the repeated multiplication underneath the notation, the zero-exponent rule stops looking like a special case too: aⁿ ÷ aⁿ obviously equals 1 (anything divided by itself is 1), and the quotient law says it should equal aⁿ⁻ⁿ = a⁰ — so a⁰ = 1 isn't a separate definition, it's forced by the quotient law being consistent with itself.

Why Power of a Power Multiplies Rather Than Adds

(aᵐ)ⁿ means aᵐ, taken as a single block, multiplied by itself n times: aᵐ × aᵐ × ... (n times). Each of those n copies of aᵐ contributes m factors of a, and there are n such copies, giving m × n factors of a in total — hence (aᵐ)ⁿ = aᵐⁿ. This is a genuinely different operation from aᵐ × aⁿ, even though both involve two exponents and one base; multiplying two separate powers together adds their exponents, but raising one power to another multiplies them, because one case is counting total copies from two separate groups, and the other is counting copies within copies.

Where a Law Gets Applied to the Wrong Part

• Adding exponents when the bases differ. The product law only adds exponents when the base is identical on both sides — 2³ × 3² cannot be simplified this way at all.
• Mishandling a negative sign inside a power. (-2)⁷ and -2⁷ are different: the first cubes (raises to the 7th power) the negative number itself, the second negates 2⁷ after computing it — though for this particular case both give the same value, they aren't the same expression in general, and mixing them up causes errors elsewhere — especially with even exponents, where (-2)⁶ is positive (64) but -2⁶ is negative (-64) — the same digits, opposite signs, from placing the parentheses differently.
• Forgetting to flip a fraction under a negative exponent. (3/4)⁻³ becomes (4/3)³, not -(3/4)³ or 3/4³ — the negative exponent inverts the whole fraction, not just its sign.

Applying the Rules in the Right Order

When an expression has several operations stacked together — a power raised to another power, then multiplied by something else, then divided — working from the innermost operation outward keeps the process manageable. Simplify any (aᵐ)ⁿ first, since power-of-a-power collapses two exponents into one before anything else needs to happen to it; only after that's done should the product or quotient laws be applied to combine the result with the rest of the expression. Trying to apply several laws simultaneously in one line is where most errors in multi-step problems creep in.

From Six Laws to Standard Form

These six laws are exactly what makes converting between ordinary numbers and exponential notation possible in Exercise 4.2, where very large and very small real-world measurements are written in standard form. The same rules reappear when simplifying algebraic expressions with variable exponents later this year.