005i - Problem Solving with Exponent Laws

Lesson 5 — Problem Solving and Consolidation: Exponent Laws

Strand: Number | Descriptor: AC9M8N02 | Duration: 45 minutes

Learning Intentions

• To flexibly apply all four exponent laws — multiplication, division, power-of-a-power, and zero-exponent — to solve applied and non-routine problems.
• To justify the choice of law(s) used and check the reasonableness of a solution.

Success Criteria

I can:

1. Choose the appropriate exponent law(s) to solve a problem, and justify the choice.
2. Solve applied problems involving powers with the same base.
3. Solve simple equations involving exponents by comparing indices.
4. Explain my reasoning clearly, including checking whether my answer is plausible.

Warmup

(6 minutes — “Always, sometimes, never”, pairs)

Decide whether each statement is always, sometimes, or never true. Give a supporting example or counterexample.

1. .
2. .
3. Increasing the exponent of a power with base greater than increases its value.
4. .

Answers: 1. Never (with the laws studied) — the multiplication law gives , not ; confusing this with the power-of-a-power law is the most common exponent error. 2. Sometimes — true for any ; is excluded/undefined at this level. 3. Always — true for any base and positive integer exponents. 4. Always — both simplify to , since multiplication of the exponents is commutative.

Activities

Activity 1 — Quick-fire Law Selection (10 min)

Explicit instruction, then practice.

Remind students of the full toolkit from Lessons 1–4:

Law	Rule
Multiplication
Division
Power of a power
Zero exponent	(for )

I do:

You do: Simplify and evaluate:

Activity 2 — Applied Problems (23 min)

Pairs. Every answer must carry a one-sentence justification of which law(s) were used.

Problem 1. A computer’s memory doubles with each generation. Generation has gigabytes. If Generation has GB and Generation has GB, how many times bigger is Generation ‘s memory than Generation ‘s?

Problem 2. A single bacterium splits into bacteria every hour. After hours there are bacteria. Find, as a single power of , the population after hours, then again after a further hours (i.e. after hours total), and use the multiplication law to explain the relationship between the two populations.

Problem 3. Solve for : .

Problem 4 (harder — applied, multi-step). A charity doubles its number of donors every year. In Year (the founding year) it had donors. By Year , the number of donors is .

(a) If and by Year the charity has donors, check whether this is consistent with the model .

(b) Suppose instead the charity’s donor count is described by , for any year . Simplify this expression. What do you notice, and what does it tell you about the growth pattern?

Socratic scaffolding for Problem 4(b):

Prompt	Purpose
Understand: what is being asked?	Simplify a fraction of powers where the same variable appears in both exponents, then interpret the result.
Devise a plan	Apply the division law: subtract the denominator’s exponent from the numerator’s exponent.
Carry out the plan	, so the expression simplifies to .
Looking back — does the answer make sense?	The cancels out entirely, leaving a constant, . This means the donor count described by this expression does not actually depend on the year — it is a fixed number, , for every year .
Extension: why might this be a useful (or suspicious) model?	A constant “growth” expression might represent a fixed ratio between two related quantities rather than genuine year-on-year growth — a useful check on whether a proposed model is realistic.

Answers: Problem 1 — times bigger. Problem 2 — after hours: ; after hours: ; since , the population after hours is times the population after hours. Problem 3 — , so . Problem 4(a) — ✓ consistent. Problem 4(b) — , a constant independent of .

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Simplify .
2. A tank’s volume is litres and a second tank’s volume is litres. How many times bigger is the first tank?
3. Solve for : .
4. Reasoning. Explain why simplifies to a constant value, no matter what positive integer is. Find that constant.

Answers: 1. ; 2. times bigger; 3. ; 4. The division law gives — the cancels because it appears identically in both exponents, leaving a fixed value of .

Common Misconceptions

Misconception	How to pre-empt it
Confusing the multiplication law with the power-of-a-power law when a variable exponent is involved, e.g. treating as needing multiplication.	Anchor every step to “am I combining two separate powers (add/subtract exponents) or raising a power to a further power (multiply exponents)?”
Assuming an expression with a variable exponent, like , cannot be simplified without knowing .	Show that the laws still apply symbolically — the variable exponent behaves exactly like a numeral throughout.
Dropping the "" when a zero-exponent term appears mid-expression, losing track of other factors.	Require the zero-exponent term to be written explicitly as for one working line before it is omitted.
Not checking whether a final answer is sensible in context (e.g. a “number of donors” coming out negative or non-integer).	Build a “looking back” step into every applied problem: does the number of items, generations, or hours make sense?
Treating “how many times bigger” as a subtraction problem rather than a division problem.	Contrast directly: “difference” language needs subtraction; “how many times” language needs division — practise identifying which is which.

Enrichment — Competition-Style Problems

E1 (AMC Junior style). If , find .

Answer

E2 (Kangaroo style). Simplify for any positive integer .

Answer

The result is a constant, , regardless of .

E3 (Challenge). Find the smallest positive integer such that .

Answer

Simplify the right-hand side first:

So the inequality becomes . Testing: (too small), . The smallest such is .

E4 (Investigation). A single fold of a piece of paper doubles its thickness. A piece of paper starts at thickness mm (i.e. mm). After how many folds does the thickness first exceed mm?

Answer

After folds, thickness mm. Testing powers of : , . So the thickness first exceeds mm after folds. (This is the classic “paper folding” growth problem — exponential growth outpaces intuition very quickly.)

Homework

1. Simplify, leaving each answer as a single power: (a) (b) (c) .
2. Solve for : (a) (b) .
3. A tank has volume litres. A second tank has volume litres. How many times bigger is the first tank than the second?
4. A student claims that gets larger as increases. Evaluate the expression and decide if the student is correct.
5. Reasoning. Explain, using the laws, why always simplifies to , no matter what value takes (with ).
6. Challenge. A colony of insects triples every week, starting from insects in week . In which week does the population first exceed insects?

Answers: 1(a) (b) (c) . 2(a) (b) . 3. times bigger. 4. The student is incorrect — the expression simplifies to , a constant, since the cancels in the division; it does not change as increases. 5. Applying the division law, ; the always cancels because it appears identically in both the numerator’s and denominator’s exponents, leaving only . 6. Population in week is . Testing: while , so the population first exceeds in week .