Exponential Functions and Exponent Rules Review – Step-by-Step Study Guidance

Exponent Rules Practice

Background

Topic: Exponent Rules

These questions test your understanding of how to simplify expressions involving exponents, including multiplication, division, and powers of powers.

Key Terms and Formulas

• Product Rule: $x^a \cdot x^b = x^{a+b}$

• Quotient Rule: $\frac{x^a}{x^b} = x^{a-b}$

• Power Rule: $(x^a)^b = x^{a \cdot b}$

• Distributive Rule for Exponents: $(ab)^n = a^n b^n$

Step-by-Step Guidance

1. Identify the base and the exponents in each expression.

2. Apply the appropriate exponent rule (product, quotient, or power) to combine or simplify the exponents.

3. For expressions with multiple variables, treat each variable separately using the rules above.

4. Check if you need to distribute exponents over multiplication or division.

Try solving on your own before revealing the answer!

Writing Exponential Functions from Tables

Background

Topic: Exponential Functions

These questions ask you to write an exponential function based on a table of values. You need to identify the initial value and the common ratio.

Key Terms and Formulas

• General form: $y = a \cdot r^x$

• $a$ = initial value (when $x = 0$)

• $r$ = common ratio (the factor by which $y$ changes as $x$ increases by 1)

Step-by-Step Guidance

1. Look at the table and identify the value of $y$ when $x = 0$; this is your initial value $a$.

2. Find the ratio between consecutive $y$ values to determine the common ratio $r$.

3. Write the function in the form $y = a \cdot r^x$.

4. Check your function by plugging in values of $x$ to see if you get the corresponding $y$ values.

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Q1. Sam took 800 mg of Ibuprofen for her nagging backache. Every hour that the Ibuprofen is in her system, the medicine’s strength decreases by 17%. How many milligrams are left in her system after 8 hours?

Background

Topic: Exponential Decay

This question tests your ability to model exponential decay, where a quantity decreases by a fixed percentage each time period.

Key Terms and Formulas

• Exponential decay formula: $A = A_0 \cdot (1 - r)^t$

• $A_0$ = initial amount (800 mg)

• $r$ = decay rate (as a decimal, 0.17)

• $t$ = number of time periods (hours)

Step-by-Step Guidance

1. Identify the initial amount: $A_0 = 800$ mg.

2. Convert the percentage decrease to a decimal: $r = 0.17$.

3. Determine the number of hours: $t = 8$.

4. Set up the formula: $A = 800 \cdot (1 - 0.17)^8$.

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Q2. The population of a city is currently 54,320 people. The population increases by 3.2% each year. Estimate the population in 2030.

Background

Topic: Exponential Growth

This question tests your ability to use exponential growth to estimate future population given a percentage increase per year.

Key Terms and Formulas

• Exponential growth formula: $P = P_0 \cdot (1 + r)^t$

• $P_0$ = initial population (54,320)

• $r$ = growth rate (as a decimal, 0.032)

• $t$ = number of years from now to 2030

Step-by-Step Guidance

1. Identify the initial population: $P_0 = 54,320$.

2. Convert the percentage increase to a decimal: $r = 0.032$.

3. Calculate the number of years between now and 2030.

4. Set up the formula: $P = 54,320 \cdot (1 + 0.032)^t$.

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Q3. You invested your allowance of $270 which gets 15% compounded weekly for 3 years. How much will you have at the end of your 3 years?

Background

Topic: Compound Interest (Exponential Growth)

This question tests your ability to use the compound interest formula to calculate the future value of an investment.

Key Terms and Formulas

• Compound interest formula: $A = P \cdot \left(1 + \frac{r}{n}\right)^{n t}$

• $P$ = principal (initial investment, $270)

• $r$ = annual interest rate (as a decimal, 0.15)

• $n$ = number of compounding periods per year (52 for weekly)

• $t$ = number of years (3)

Step-by-Step Guidance

1. Identify the principal: $P = 270$.

2. Convert the interest rate to a decimal: $r = 0.15$.

3. Determine the number of compounding periods per year: $n = 52$.

4. Set up the formula: $A = 270 \cdot \left(1 + \frac{0.15}{52}\right)^{52 \cdot 3}$.

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Q4. You are going to buy your first car! You decide on buying a used Honda Accord for $23,500. You decide to finance your car and take out a loan. Look at the different banks and determine which would be the best bank to go with for your loan!

Background

Topic: Compound Interest and Loan Comparison

This question tests your ability to compare different loan offers using compound interest formulas to determine which is more cost-effective.

Key Terms and Formulas

• Compound interest formula: $A = P \cdot \left(1 + \frac{r}{n}\right)^{n t}$

• $P$ = principal (loan amount, $23,500)

• $r$ = annual interest rate (as a decimal)

• $n$ = number of compounding periods per year (1 for annually, 12 for monthly)

• $t$ = loan term in years

Step-by-Step Guidance

1. For each bank, identify the interest rate, compounding frequency, and loan term.

2. Convert the interest rates to decimals: 8.5% = 0.085, 1.9% = 0.019.

3. Set up the compound interest formula for each bank:

   • CMWFCU: $A = 23,500 \cdot (1 + 0.085)^5$

   • TrustMe Bank: $A = 23,500 \cdot \left(1 + \frac{0.019}{12}\right)^{12 \cdot 3}$

4. Calculate the total amount owed for each bank (stop before the final calculation).

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