Transform exponential expressions with the properties of exponents to reveal growth and decay rates

Learning goal: Use the properties of exponents to transform expressions for exponential functions, choosing and producing an equivalent form that reveals and explains properties of the quantity represented, such as the rate of growth or decay per year, per month or per day.

The properties in play. For a positive base b: the power rule (b^m)^n = b^(mn), the product rule b^m · b^n = b^(m+n), and the product-to-a-power rule (ab)^n = a^n b^n, together with the meaning of rational exponents, b^(1/n) = ⁿ√b. Each property lets an exponential expression be rewritten without changing its value, and each rewriting exposes a different property of the quantity. The example in the standard is the simplest case: 8^t can be written as (2³)^t = 2^(3t). Read the two forms. The first says "multiply by 8 each unit of t"; the second says "multiply by 2 three times each unit of t", which is the same thing seen at a finer scale. The transformed form reveals that doubling happens three times per unit.

Changing the time unit reveals a different rate. Suppose a quantity grows by 15% per year, so it is multiplied by 1.15 each year and after t years has the factor 1.15^t. What is the equivalent monthly growth rate? There are 12 months in a year, so write t years as 12t months and use the power rule in reverse: 1.15^t = 1.15^((1/12)·12t) = (1.15^(1/12))^(12t). The new base, 1.15^(1/12), is the twelfth root of 1.15, approximately 1.011715. So the expression is approximately 1.011715^(12t), and its reading is "multiply by 1.011715 each month for 12t months". The monthly growth rate is about 1.17%, not 15/12 = 1.25%. The difference is compounding: twelve monthly increases of 1.25% would give 1.0125^12 ≈ 1.1608, a 16.08% annual increase, more than 15%. The exponent property gives the exact conversion; dividing the percent by 12 does not. Check: 1.011715^12 ≈ 1.1500. The exact form is 1.15^(1/12); the decimal is an approximation and should be labeled as one.

A worked model inside the lesson, decay. A hypothetical substance loses half its mass every 8 days, so its mass after t days is M = M₀ · (1/2)^(t/8). What is the daily decay rate? Rewrite with the power rule: (1/2)^(t/8) = ((1/2)^(1/8))^t. The daily factor is (1/2)^(1/8), the eighth root of 1/2, approximately 0.9170. So M ≈ M₀ · 0.9170^t, and the mass shrinks by about 1 − 0.9170 = 0.0830, that is about 8.3% per day. Check: 0.9170⁸ ≈ 0.5000. The original form reveals the halving time; the transformed form reveals the daily rate. Both describe the same quantity, and a question about "per day" is answered by producing the second.

Going the other way: from a short period to a long one. A balance grows by 0.25% per month, so it is multiplied by 1.0025 each month and by 1.0025^(12t) over t years. Rewrite: 1.0025^(12t) = (1.0025^12)^t ≈ 1.03042^t. The annual growth factor is about 1.0304, an effective annual rate of about 3.04%, slightly more than 12 × 0.25% = 3% because each month's growth earns growth in later months. The nominal rate 3% and the effective rate 3.04% are different numbers, and the exponent property is what converts one to the other.

Shifting the exponent reveals a starting value. The product rule handles sums in the exponent: 4^(x+1) = 4^x · 4^1 = 4 · 4^x, and 2^(x+3) = 8 · 2^x. So an expression such as 50 · 2^(x+3) is the same as 400 · 2^x: the "+3" in the exponent is really a factor of 8 on the initial value. A population modeled by 300 · 2^((t−5)/10) can be read as 300 · 2^(t/10) · 2^(−1/2) = (300/√2) · 2^(t/10) ≈ 212.1 · 2^(t/10), revealing that the value at t = 0 is about 212, not 300; the 300 is the value at t = 5.

Doubling time to yearly rate. A quantity that doubles every 10 years has factor 2^(t/10) = (2^(1/10))^t ≈ 1.07177^t, so it grows about 7.18% per year. Conversely, a quantity growing 7.18% per year doubles in about 10 years. The rewriting connects two descriptions that people use interchangeably in conversation and that students often cannot connect on paper.

Conditions and cautions. These transformations require a positive base, which every growth or decay factor is. The power rule (b^m)^n = b^(mn) can rewrite one expression in many equal ways: 1.15^t also equals (1.15^12)^(t/12), which is true but reveals a 12-year factor that nobody asked for. The useful rewriting is the one whose new base belongs to the period you are asked about. Keep exact forms such as 1.15^(1/12) in the algebra and round only when reporting a rate, stating the rounding. And the rewritten rate is a per-period multiplier; a growth rate of 1.17% per month means the multiplier is 1.0117, and the 1 must be subtracted to state the percent.

Why the transformed form is worth producing. Two models with the same growth over a year can be compared only when both are written with the same period. A savings account quoted at 3% per year and one quoted at 0.25% per month can be compared once 1.0025^12 ≈ 1.0304 is computed. Writing an equivalent form is not decoration; it is the act that makes a property of the quantity, its rate per chosen period, visible and comparable.

Worked rewrite of a power of 8: Write 8^t as a power of 2 and interpret. 8 = 2³, so 8^t = (2³)^t = 2^(3t). A quantity multiplied by 8 each unit of time is doubled three times each unit of time. Check with t = 2: 8² = 64 and 2⁶ = 64. The same idea converts 9^x to 3^(2x) and 25^(x/2) to 5^x, since 25^(x/2) = (5²)^(x/2) = 5^x.

Worked annual to monthly rate: A balance grows by 15% per year: factor 1.15^t for t years. Monthly: 1.15^t = (1.15^(1/12))^(12t). The monthly factor is 1.15^(1/12) ≈ 1.011715, so the monthly rate is about 1.17%. Check: 1.011715^12 ≈ 1.1500. Compare with the naive 15% ÷ 12 = 1.25%: 1.0125^12 ≈ 1.1608, which would be 16.08% per year, too high. The exact monthly factor is the twelfth root of 1.15; 1.17% is a rounded report.

Worked half-life to daily rate: A hypothetical substance has mass M₀(1/2)^(t/8) after t days, halving every 8 days. Rewrite: (1/2)^(t/8) = ((1/2)^(1/8))^t ≈ 0.9170^t. The daily factor is about 0.9170, so the mass falls by about 8.3% per day. Check: 0.9170⁸ ≈ 0.500. After 20 days the two forms agree: (1/2)^(20/8) = (1/2)^2.5 ≈ 0.1768 and 0.9170^20 ≈ 0.1768.

Worked monthly to annual rate: A balance is multiplied by 1.0025 each month (0.25% per month). Over t years: 1.0025^(12t) = (1.0025^12)^t ≈ 1.0304^t. The effective annual rate is about 3.04%, more than 12 × 0.25% = 3.00% because monthly growth compounds. Over 5 years, 1.0025^60 ≈ 1.1616 and 1.0304^5 ≈ 1.1615, agreeing to three decimal places; the small gap comes from rounding the base to 1.0304, and the exact base 1.0025^12 reproduces 1.1616.

Equivalent-forms record: Make a table with columns given expression, property used, equivalent form, and what the new form reveals. Fill it for 8^t, 1.15^t rewritten per month, (1/2)^(t/8) rewritten per day, 1.0025^(12t) rewritten per year, 4^(x+1), 50 · 2^(x+3), and 2^(t/10) rewritten per year. For each row give the exact new base (as a root or power) and a decimal approximation to four places, and check one numerical value of the variable in both forms.

Independent performance: Choose two growth or decay situations stated with different time units, for example a population growing 4% per year and one growing 0.3% per month, or a medication losing half its amount every 6 hours and one losing 10% per hour. Write an exponential expression for each, then use the properties of exponents to rewrite both with the same period so they can be compared. State which grows or decays faster and by how much per period. Verify each rewriting by evaluating the original and the transformed expression at one value of the variable. Success criteria: each transformation names the property of exponents used, each new base is given exactly and as a rounded decimal, percentages are reported as rate = factor − 1, and the comparison is a sentence about the situations.