Exponential Growth

Grade 9 · algebra · 90 practice problems · read aloud

🔊 Listen to this explanation

Exponential Growth

📈 What is Exponential Growth?

Exponential growth happens when a quantity increases by a fixed percentage (the growth rate) over a fixed period of time. Unlike linear growth (adding the same amount), exponential growth means multiplying by the same factor. It's useful for modeling things like population growth, compound interest, and the spread of a virus.

🧮 The Formula

The standard exponential growth formula is:

y = a(1 + r)t

  • y: Final amount
  • a: Initial amount
  • r: Growth rate (as a decimal)
  • t: Number of time periods

🔢 Step-by-Step Guide

  1. Identify the initial amount (a), growth rate (r), and time (t).
  2. Convert the percentage growth rate to a decimal (divide by 100).
  3. Add 1 to the decimal growth rate to get the growth factor (1 + r).
  4. Raise the growth factor to the power of 't'.
  5. Multiply the result by the initial amount 'a' to find the final amount 'y'.

💡 Visual Examples

Example 1: Population

A town of 2,000 people grows at 5% per year. What's the population after 3 years?

  1. a = 2000, r = 5% = 0.05, t = 3
  2. Growth Factor = 1 + 0.05 = 1.05
  3. y = 2000 * (1.05)3
  4. y = 2000 * 1.157625
  5. y = 2,315 people (after rounding)

Example 2: Money

You invest $500 in a savings account with 3% annual interest. How much will you have after 4 years?

  1. a = 500, r = 3% = 0.03, t = 4
  2. Growth Factor = 1 + 0.03 = 1.03
  3. y = 500 * (1.03)4
  4. y = 500 * 1.1255
  5. y = $562.75

⚠️ Common Mistakes

  • Forgetting to convert % to decimal: 5% is 0.05, not 5 in the formula!
  • Adding instead of multiplying: It's not y = a + r*t. You must use the exponent.
  • Misapplying the exponent: (1 + r)t means multiply (1+r) by itself 't' times.

🌟 Tips & Tricks

  • Memory Aid: "Percent to Decimal, Add the One, Power Up, Multiply to Done!"
  • If the growth rate doubles (e.g., from 2% to 4%), the result grows MUCH faster than double.
  • Use a calculator for the exponent step to avoid arithmetic errors.

🎯 Practice Suggestions

To master exponential growth:

  1. Start with simple rates (2%, 5%, 10%) and small time periods (2-5 years).
  2. Create a table showing the growth year-by-year to see the pattern.
  3. Use real-world contexts you care about, like growth of social media followers or video game currency.
  4. Practice writing the formula from a word problem before plugging in numbers.

Practice problems

6 of the 90, worked through step by step — try them before opening the answer.

1 2^5 - 3^2 = ?

Hint: Evaluate each exponential term separately before performing the subtraction operation. For example, if you had 4^2 - 2^3, you would first calculate 4 squared and 2 cubed.

Show the answer

Answer: 23

  1. Calculate 2^5 2^5 means 2 multiplied by itself 5 times: 2 × 2 = 4 4 × 2 = 8 8 × 2 = 16 16 × 2 = 32 So, 2^5 = 32.
  2. Calculate 3^2 3^2 means 3 multiplied by itself 2 times: 3 × 3 = 9 So, 3^2 = 9.
  3. Subtract the results We have 2^5 - 3^2 = 32 - 9. 32 - 9 = 23. Final Answer: 23

Let's solve the problem step by step.

2 2^5 - 3^2 × 2 = ?

Hint: Remember to follow the order of operations when evaluating expressions with exponents and multiplication.

Show the answer

Answer: 14

  1. Calculate exponents first** 2^5 = 2 × 2 × 2 × 2 × 2 = 32 3^2 = 3 × 3 = 9 So the expression becomes: 32 - 9 × 2 --- **
  2. Perform multiplication before subtraction (order of operations: PEMDAS/BODMAS)** 9 × 2 = 18 Now we have: 32 - 18 --- **
  3. Perform subtraction** 32 - 18 = 14 --- **Final Answer:** 14

Let's solve step by step. We have: 2^5 - 3^2 × 2 --- **

3 2^4 × 3^2 ÷ 6 = ?

Hint: Remember to apply the order of operations and simplify exponents before multiplying and dividing.

Show the answer

Answer: 24

  1. Calculate the exponents first: 2^4 = 16 and 3^2 = 9
  2. Multiply the results: 16 × 9 = 144
  3. Divide by 6: 144 ÷ 6 = 24

The answer is 24.

4 2^3 × 3^2 ÷ 6 = ?

Hint: Evaluate each exponential term separately before performing the multiplication and division operations

Show the answer

Answer: 12

  1. Evaluate 2^3 = 2 × 2 × 2 = 8
  2. Evaluate 3^2 = 3 × 3 = 9
  3. Multiply the results: 8 × 9 = 72
  4. Divide by 6: 72 ÷ 6 = 12

The answer is 12.

5 2^(x+1) - 2^x = 16

Hint: Use exponent rules to factor the expression and solve for the variable

Show the answer

Answer: 4

  1. Factor out 2^x from the left side: 2^x(2^1 - 1) = 16
  2. Simplify inside parentheses: 2^x(2 - 1) = 16
  3. 2^x(1) = 16
  4. 2^x = 16
  5. Write 16 as a power of 2: 16 = 2^4
  6. Therefore, x = 4

The answer is 4.

6 2^(x+1) - 2^x = 32

Hint: Try factoring out the common exponential term and simplifying the equation

Show the answer

Answer: 5

  1. Factor out 2^x from the left side: 2^x(2^1 - 1) = 32
  2. Simplify inside parentheses: 2^x(2 - 1) = 32
  3. 2^x(1) = 32
  4. 2^x = 32
  5. Write 32 as a power of 2: 32 = 2^5
  6. Therefore, 2^x = 2^5
  7. Since the bases are equal, the exponents must be equal: x = 5

The answer is 5.

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