# Original model cards · Year 9 exponent laws

All domains, settings, quantities and quotes below are **fictional classroom models**. The cards deliberately move from the opening fortnight's coordinate/linear models to Week 3 numerical exponent laws and Week 4 variables in exponents. A count model uses non-negative whole numbers unless a card says otherwise; a reciprocal or quotient requires a non-zero base. A calculator may verify an arithmetic result, but the learner must show **why** the exponent rule applies. No card proves a real technology specification, measurement, safety property or price. An exact adult reading is listening-supported access, not independent print reading.

## Card A Nested message choices

A fictional game editor gives a player three independent yes/no choices in one stage and four independent yes/no choices in a second stage. The first stage has `2^3=8` possible choice strings; the second has `2^4=16`. Each first string can pair with each second string, so there are `2^3×2^4=2^(3+4)=2^7=128` combined strings. This counts **model strings**, not whether players want or can use all of them. Adding 8+16 would count the two lists separately rather than all pairs. Counting assumes each yes/no option is available independently.

## Card B Grouping a paper-cell array

An invented square-paper array has `3^5=243` equal cells. The editor divides it into groups of `3^2=9` cells without leftovers: `3^5÷3^2=3^(5−2)=3^3=27` groups. The quotient says **groups of cells**, not a new physical sheet size. Division by a base-zero power is not permitted. A separate dimensionless ratio of equal non-zero counts, `3^2÷3^2=3^0=1`, explains why this non-zero base to exponent zero equals 1.

## Card C A small paper scale

For a **drawn** length model, a scale factor `10^−2=1/10^2=1/100` multiplies an invented 1 m strip to give `0.01 m = 1 cm`. The factor is dimensionless; the result is a length. The [BIPM SI-prefix table](https://www.bipm.org/en/measurement-units/si-prefixes) identifies centi- as `10^−2`. No real object or site was measured. A negative exponent does **not** make the length negative, and `10^−2` is **not** `−100`.

## Card D Cue combinations

A fictional stage notebook has `2^3=8` possible paper cue cards. An ordered pair of cards has `(2^3)^2=2^(3×2)=2^6=64` model pairs. This is different from `2^3×2^2=2^5=32`: the second expression gives one eight-choice stage paired with one four-choice stage. The bracket changes the repeated factor. No actual lighting or stage device is being operated.

## Card E Variable choices in independent slots

Let `x` be a positive whole number of abstract choices at **each** independent slot. Three slots have `x^3` choice strings; two more slots have `x^2`. Together, `x^3×x^2=x^5`. If `x=2`, the model has `2^5=32` strings. If `x=0`, the count context has no available choice, but the **quotient** `x^5÷x^2=x^3` would be undefined because it divides by zero; that identity requires `x≠0`. The symbolic product and quotient have different domain requirements.

## Card F A non-zero scale ratio

Let `u` be a non-zero real, unitless scale factor. The ratio `u^2÷u^5=u^(2−5)=u^−3=1/u^3`. At `u=2`, the result is `1/8`. At `u=−2`, it is `−1/8` because an odd power preserves the negative sign. At `u=0`, the original quotient and reciprocal are undefined. This is a **ratio**, not a count of objects or a negative physical length.

## Card G A variable exponent in a game rule

For a non-negative whole stage number `n`, a fictional puzzle maker uses `2^n` first-stage paths and `2^(n+2)` second-stage paths, independently. Total model path pairs are `2^n×2^(n+2)=2^(2n+2)`. At `n=2`, this is `2^2×2^4=4×16=64=2^6`. The exponents add as algebraic expressions. This does not assert a game has been built or that all paths are usable.

## Card H Poster icon grid

For a non-negative whole `r`, a fictional design sketch has `2^r` paper icons in each row and the same number of rows. The model total is `(2^r)^2=2^(2r)` icons, not `2^(r+2)`. At `r=3`, eight rows of eight make 64 icons. The sketch says nothing about print clarity or available wall space. If a different design has only two rows of `2^r` icons, its total is `2×2^r=2^(r+1)`, a different condition.

## Shared rule boundary

For non-zero base `a` and integers `m,n`: `a^m a^n=a^(m+n)`, `a^m/a^n=a^(m−n)`, `(a^m)^n=a^(mn)`, `a^0=1`, `a^(−n)=1/a^n`. These are **same-base product/quotient** and power-of-power rules, not rules for sums: `a^m+a^n` cannot generally be simplified to `a^(m+n)`. For negative integer exponents or a quotient, the base must be non-zero. Parentheses control whether a negative sign is part of the base: `(-2)^4=16`, whereas `-2^4=−16` under the usual order of operations. The [teacher key](TEACHER-KEY.md) holds feedback examples; fresh check numbers are separate.

**Original resource rights:** © NeuroForgeIO Pty Ltd 2026, SubjectNest, [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). Credit author, source, licence and changes. SI-prefix source credited above; no external text/artwork has been copied into these cards.