Use exponent laws without losing the base conditions
Compare multiplicative change, equivalent exponential forms and exact equations.
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Two original synthetic populations both start at 100. Model A adds 20 each period. Model B multiplies its current amount by 1.2 each period. They agree after the first step. Predict whether they still agree after the second and third steps.
The formulas are A(t)=100+20t and B(t)=100(1.2)^t for nonnegative whole-number periods t. Here the base 1.2 is a retained multiplier, not an amount added. The choice of a discrete time domain is part of this first model.
Predict
Predict what a 20% reduction would do. It multiplies the current amount by 0.8, not by −0.2 and not by 0.2. A percentage change refers to a denominator: here each period’s starting amount. Repeated equal percentage changes therefore produce unequal absolute changes.
Experiment
Build and compare a table.
t
Additive A(t)
Multiplicative B(t)
Change in B from previous period
0
100
100
Not applicable
1
120
120
20
2
140
144
24
3
160
172.8
28.8
Plot the two models on shared axes. Equal input steps create a constant difference for A and a constant ratio for B. Two matching rows do not establish that the models are the same. Now repeat with multiplier 0.8 and explain why the amount remains positive while decreasing.
Understand
For positive base b, b^u b^v=b^(u+v), b^u/b^v=b^(u−v), and (b^u)^v=b^(uv) for real exponents. With integer exponents, some laws also extend to negative bases, but rational or arbitrary real powers need more careful domain choices. Keeping the base positive is a safe foundation for the exponential family studied here.
For a nonzero base, b⁻ⁿ=1/bⁿ. A negative exponent changes multiplication to reciprocal; it does not usually create a negative output. Also b⁰=1. Expressions involving a zero base require separate checks, particularly negative exponents and 0⁰, which is not assigned a value by these rules.
For b>0 and b≠1, the real function b^x is one-to-one. Thus b^u=b^v implies u=v. Solve 4^(x−1)=8 by rewriting both sides using base 2: 2^(2x−2)=2³, so 2x−2=3 and x=5/2. Matching exponents would not work with base 1, whose every output is 1.
In N(t)=Ab^t, A is the value at t=0 and b is the multiplier for one time unit. Two positive observations at distinct times determine a positive b within this assumed model. If N(2)=45 and N(4)=180, their ratio gives b²=4, so b=2 and A=45/4. The positive-base condition selects the relevant root.
Fitting is not validation. A model can match two measurements exactly and fail later because of resource limits, changing conditions or measurement error. Context also determines whether fractional times are meaningful and whether outputs should be rounded. Extending a discrete formula to continuous time is an additional modeling choice.
Apply
The practice asks for exact exponent arithmetic, an equation, a synthetic model fit and a false-law diagnosis. Explain why a product law cannot be transferred to a sum. For a model, state the time unit, initial value and multiplier before predicting.
The compound-growth experiment lets you explore repeated multiplication. Treat its teaching assumptions as assumptions rather than evidence about an actual investment or population.
Continue
When a target cannot conveniently be expressed as a power of the same base, logarithms recover the exponent. Their domain restrictions are just as important as their algebraic laws.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Evaluate 2⁻³ × 2⁵ exactly.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Solve 3^(2x−1)=27.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Evaluate 16^(−3/4) exactly using the positive real fourth root.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
Which formula starts at 240 and loses 15% of its current amount each period?
An original synthetic culture model N(t)=Ab^t has N(1)=30 and N(3)=120, with A>0 and b>0. Find A and b, then predict N(4). State what two observations cannot establish about a real culture.
Structure your argument and preserve revisions
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Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Disprove (a+b)²=a²+b² with positive numbers. Then explain why (ab)²=a²b² is valid and why 0⁻¹ cannot be evaluated by the negative-exponent rule.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
