Equivalent Exponential Forms and Different Time Intervals

8.6: Equivalent Exponential Forms and Different Time Intervals. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 6 of 63 parts Practice Flashcards

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Convert to a longer interval

Objective: Raise a short-interval factor to the number of intervals.

Prerequisite skills: rational exponents and exponential factors. If the factor per unit is b, the factor for k such units is b^k. This is repeated multiplication, not k times the growth rate unless an approximation is explicitly allowed. Common error: multiplying b by the number of periods. Restrictions and interpretation: Changing the time unit changes the factor and exponent together; do not divide a factor by the interval length.

Formulas

k-period factor=b^k

Worked examples

Try itA quantity multiplies by 3.2 each day. What is the exact multiplicative factor over 2 days?Show answer

Two daily intervals apply the factor twice: 3.2×3.2=10.24. This is b², not 2b. Therefore the requested answer is 10.24.

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Convert to a longer interval

Objective: Raise a short-interval factor to the number of intervals.

Formulas

k-period factor=b^k

Worked examples

Try itA quantity multiplies by 3.2 each day. What is the exact multiplicative factor over 2 days?Show answer

Two daily intervals apply the factor twice: 3.2×3.2=10.24. This is b², not 2b. Therefore the requested answer is 10.24.

Convert to a shorter interval

Objective: Take a root of a longer-interval factor.

If a factor B applies over k equal units, the constant one-unit factor is the positive kth root B^(1/k). Raising that factor to k must reproduce B. Percent rates require subtracting 1 afterward. Common error: dividing B by k instead of taking a root. Restrictions and interpretation: Changing the time unit changes the factor and exponent together; do not divide a factor by the interval length.

Formulas

unit factor=B^(1/k)

Worked examples

Try itA quantity is multiplied by 13824 every 3 years under an exponential model. What is the yearly multiplicative factor?Show answer

Let g be the positive yearly factor. Then g³=13824=24³, so g=24. Multiplying by 24 three times recovers 13824. Therefore the requested answer is 24.

Equivalent time-unit representations

Objective: Adjust the exponent when the input time unit changes.

The same process can be expressed in days, months, or years only when conversion factors are included in the exponent. Let t count new units, express the old elapsed time in t, then substitute. Common error: changing the time label without changing the exponential rate. Restrictions and interpretation: Changing the time unit changes the factor and exponent together; do not divide a factor by the interval length. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$A·B^(t/k)=A·(B^(1/k))^t$

Worked examples

Try itA model is P(d) = 27·4^(d/6), with d in days. What is P(9)?Show answer

At d=9, exponent 9/6=3/2. Since 4^(3/2)=(√4)³=8, P(9)=27×8=216. An equivalent daily factor is 4^(1/6). Therefore the requested answer is 216.

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