Chapter 11.2 — Exercise 11.1 — Monomial x Monomial

Running Five Short Pairs Through the Same Pattern

The third and fifth rows are worth a second look, since they're the two that combine a repeated variable rather than two different ones: t² × t² adds exponents to give t⁴ (not t², a common slip), and p² × p gives p³ (2 + 1, not 2 × 1).

A Small Multiplication Table, Built From Scratch

The same idea scales up into a grid: pick a handful of row-monomials and column-monomials, and fill in every cell with that row times that column.

A grid like this is really nine of the pair-products from the section above, just arranged so every row-monomial gets multiplied against every column-monomial systematically — the exact task Exercise 11.1 sets out for a larger 8×8 version of the same idea.

Volumes Are Just Three Monomials Multiplied Together

Length × breadth × height, applied to monomial dimensions, is nothing more than the two-monomial rule run twice in a row:

The last row is the busiest, with all three of p, q, and r spread across the three dimensions — multiplying the numbers (3 × 2 × 4 = 24) and then each variable's exponents separately (p¹×p¹=p², q¹×q¹=q², r¹×r¹=r²) gives 24p²q²r², even though no single dimension contains all three letters on its own. A box like this could just as easily describe a real container whose length, breadth, and height each happen to scale with a different quantity — p bricks laid one way, q laid another, r stacked in height — with the algebra keeping track of exactly how the total volume scales as any one of those quantities changes, without needing to plug in actual numbers first.

Chaining More Than Two Monomials

Nothing about the rule changes when three, four, or five monomials are multiplied in one go — every number multiplies together, and every repeated variable's exponents add up, regardless of how many separate factors they're spread across:

The last one is a useful reminder about signs: three negative-or-positive factors multiply exactly like ordinary numbers do — here −3 and −6 are both negative, and a negative times a negative is positive, so the two negative signs cancel to leave a positive 72, even before any of the letters get combined.

Checking a Long Product Using Degree Alone

With four or five monomials multiplied together, it's easy to lose track of an exponent partway through. A quick sanity check, borrowed from the introduction to this chapter, is to add up every factor's degree and confirm it matches the degree of the final answer. Take a × b × ab × a³b × ab³: the five factors have degrees 1, 1, 2, 4, and 4 respectively, adding to 12. The final answer, a⁶b⁶, has degree 6 + 6 = 12 — an exact match, confirming no exponent was dropped or miscounted along the way. Running the same check on kl × lm × km × klm: degrees 2, 2, 2, 3 add to 9, and k³l³m³ has degree 3+3+3 = 9, matching again. This check won't catch every possible mistake — it can't tell k³l³m³ apart from, say, k²l⁴m³, since both happen to have degree 9 — but it does catch the single most common error, an exponent that got added incorrectly or a variable that got left out entirely.

Letters Standing in for Whole Monomials

The same rule applies just as cleanly when the monomials themselves are given short names. If A = xy, B = yz, and C = zx, then:

And if P = 4x², T = 5x, and R = 5y, then PTR ÷ 100 works out to a clean result once the numbers are gathered separately from the letters:

Naming a monomial with a capital letter doesn't change anything about how it multiplies — P, T, and R behave exactly like 4x², 5x, and 5y would if written out in full, right up until the final division tidies the answer into a much simpler form than PTR alone would suggest.

Choosing Your Own Monomials

The exercise's final question flips the task around entirely: instead of multiplying given monomials, pick some of your own and multiply those. Three monomials chosen freely — say 2a, 3a²b, and −4b³ — multiply exactly the same way as any of the examples above:

There's nothing special about this particular choice of three monomials — any other set would multiply out just as predictably, which is really the point of the exercise: by this stage, monomial multiplication should work the same way regardless of which specific letters or numbers happen to be involved, whether they were handed to you in a textbook or picked at random.

Why the Order of Multiplying Never Matters

Every example above quietly relies on one fact that's worth stating outright: multiplication of numbers and variables can be reordered and regrouped freely without changing the result. In −2a²b × 4a × 8b² from the introduction, the three numbers (−2, 4, 8) were gathered together, then the a-powers (a² and a), then the b-powers (b and b²) — even though the original expression interleaves numbers and letters in a different order entirely. This freedom to regroup is exactly why the "multiply the numbers, then combine each variable separately" strategy is valid in the first place; without it, every product would need to be worked through strictly left to right, term by term, with no shortcut available at all. It's also why the multiplication table earlier in this lesson doesn't care whether a monomial sits in the row position or the column position — 2x × 5x and 5x × 2x are the same product, 10x², arrived at either way.

From Monomials to Full Polynomials

Every product on this page involved multiplying single terms against single terms. Exercise 11.2 extends the same underlying rule to a monomial multiplied by an entire binomial or trinomial — the calculation just gets applied once for every term inside the bracket instead of once overall, so nothing learned in this lesson goes to waste — it becomes the inner step repeated inside every larger product from here on.