Unit 6 Exponents And Exponential Functions Homework 4 Answer Key

It sounds like you're looking for the answer key to a specific homework assignment on exponents and exponential functions, likely related to a math course. While I can't provide direct answer keys (that wouldn't be very helpful for learning!), I can give you a comprehensive explanation of the concepts covered in Unit 6, along with examples and problem-solving strategies that will equip you to tackle your homework successfully. Think of this as your ultimate guide to mastering exponents and exponential functions!

Understanding Exponents: The Foundation

Exponents are a shorthand way of representing repeated multiplication. The expression bⁿ means you multiply the base b by itself n times. b is the base, and n is the exponent (or power).

• Example: 2³ = 2 * 2 * 2 = 8

Key Exponent Rules to Remember

Mastering exponent rules is crucial for simplifying expressions and solving equations. Here's a breakdown of the most important ones:

1. Product of Powers: When multiplying exponents with the same base, add the exponents.

   • b^m * bⁿ = b^m+n
   • Example: 3² * 3⁴ = 3²⁺⁴ = 3⁶ = 729

2. Quotient of Powers: When dividing exponents with the same base, subtract the exponents.

   • b^m / bⁿ = b^m-n
   • Example: 5⁵ / 5² = 5^5-2 = 5³ = 125

3. Power of a Power: When raising a power to another power, multiply the exponents.

   • (b^m)ⁿ = b^mn*
   • Example: (2³)² = 2^3*2 = 2⁶ = 64

4. Power of a Product: The power of a product is the product of the powers.

   • (ab)ⁿ = aⁿbⁿ
   • Example: (4x)² = 4² * x² = 16x²

5. Power of a Quotient: The power of a quotient is the quotient of the powers.

   • (a/b)ⁿ = aⁿ/bⁿ
   • Example: (x/3)³ = x³ / 3³ = x³ / 27

6. Zero Exponent: Any non-zero number raised to the power of zero is 1.

   • b⁰ = 1 (where b ≠ 0)
   • Example: 7⁰ = 1

7. Negative Exponent: A negative exponent indicates a reciprocal.

   • b^-n = 1/bⁿ
   • Example: 2^-3 = 1/2³ = 1/8

8. Fractional Exponent: A fractional exponent represents a root. b^m/n is the nth root of b^m.

   • b^m/n = ⁿ√(b^m)
   • Example: 4^1/2 = √4 = 2

Practice Problems with Exponent Rules

Let's solidify your understanding with some practice problems:

1. Simplify: (x³y²)⁴

   • Applying the power of a product and power of a power rules: x³⁴y²⁴ = x¹²y⁸

2. Simplify: (12a⁵b³) / (4a²b)

   • Dividing coefficients and applying the quotient of powers rule: (12/4) * a^5-2 * b^3-1 = 3a³b²

3. Simplify: (5^-2 * 5⁴) / 5⁰

   • Applying the product of powers, zero exponent, and then quotient of powers rules: 5^-2+4 / 1 = 5² = 25

4. Simplify: 16^3/4

   • Rewrite as a radical: ⁴√(16³) = ⁴√(4096) = 8

Exponential Functions: Growth and Decay

Now, let's move on to exponential functions. An exponential function is a function where the independent variable (usually x) appears in the exponent. The general form is:

• f(x) = ab^x

  • a is the initial value (the value of the function when x = 0).
  • b is the base, and it determines whether the function represents growth or decay.
  • x is the independent variable.

Understanding Growth and Decay

The base b plays a crucial role in determining the behavior of the exponential function:

• Exponential Growth: If b > 1, the function represents exponential growth. As x increases, f(x) increases rapidly.
• Exponential Decay: If 0 < b < 1, the function represents exponential decay. As x increases, f(x) decreases rapidly, approaching zero.

Graphing Exponential Functions

To graph an exponential function, follow these steps:

1. Create a table of values: Choose a few values for x (both positive and negative) and calculate the corresponding values of f(x).
2. Plot the points: Plot the points from your table on a coordinate plane.
3. Draw the curve: Connect the points with a smooth curve. Remember that exponential functions have a horizontal asymptote, which is the x-axis (y = 0) for basic exponential functions.

Key Characteristics of Exponential Function Graphs:

• They pass through the point (0, a), where a is the initial value.
• They have a horizontal asymptote at y = 0 (unless the function is shifted vertically).
• They either increase (growth) or decrease (decay) rapidly.
• They are not linear (they don't form a straight line).

Applications of Exponential Functions

Exponential functions are used to model a wide variety of real-world phenomena, including:

• Population Growth: The growth of a population (of people, animals, or bacteria) can often be modeled using an exponential function.
• Compound Interest: The amount of money in an account that earns compound interest grows exponentially.
• Radioactive Decay: The decay of radioactive substances follows an exponential decay model.
• Spread of Diseases: The spread of a disease can sometimes be modeled using an exponential function, at least in the early stages.
• Cooling/Heating: Newton's Law of Cooling describes how the temperature of an object changes exponentially as it approaches the ambient temperature.

Examples of Exponential Function Problems

Let's look at some examples of problems involving exponential functions:

1. Population Growth: The population of a town is currently 10,000 and is growing at a rate of 5% per year. Write an exponential function to model the population after t years.

   • f(t) = 10000(1 + 0.05)^t = 10000(1.05)^t

2. Compound Interest: You invest $5,000 in an account that earns 3% interest compounded annually. How much will you have after 10 years?

   • f(t) = 5000(1 + 0.03)¹⁰ = 5000(1.03)¹⁰ ≈ $6,719.58

3. Radioactive Decay: A radioactive substance has a half-life of 20 years. If you start with 100 grams, how much will remain after 60 years?

   • After 20 years (1 half-life): 50 grams
   • After 40 years (2 half-lives): 25 grams
   • After 60 years (3 half-lives): 12.5 grams
   • Alternatively, you can use the formula: f(t) = 100(1/2)^t/20 = 100(1/2)^60/20 = 100(1/2)³ = 12.5 grams

Solving Exponential Equations

An exponential equation is an equation where the variable appears in the exponent. Here are some common techniques for solving exponential equations:

1. Equating the Bases: If you can rewrite both sides of the equation with the same base, then you can equate the exponents.

   • Example: Solve 2^x = 8

     • Rewrite 8 as 2³: 2^x = 2³
     • Since the bases are the same, equate the exponents: x = 3

2. Using Logarithms: If you can't easily equate the bases, you can use logarithms to solve for the variable. The logarithm function is the inverse of the exponential function.

   • Example: Solve 5^x = 20

     • Take the logarithm of both sides (you can use any base, but the common logarithm (base 10) or the natural logarithm (base e) are most convenient): log(5^x) = log(20)
     • Use the power rule of logarithms (log(a^b) = blog(a)): xlog(5) = log(20)
     • Solve for x: x = log(20) / log(5) ≈ 1.86

3. Using Properties of Exponents to Simplify: Sometimes, you can simplify the equation using exponent rules before applying logarithms.

   • Example: Solve 4^x+1 = 16^x

     • Rewrite 16 as 4²: 4^x+1 = (4²)^x
     • Simplify: 4^x+1 = 4^2x
     • Equate exponents: x + 1 = 2x
     • Solve for x: x = 1

Logarithms: A Quick Review

Since logarithms are essential for solving many exponential equations, let's review some key concepts:

• Definition: The logarithm of a number y to the base b is the exponent to which b must be raised to produce y.

  • log_b(y) = x <=> b^x = y

• Common Logarithm: The common logarithm (log) has a base of 10. log(y) = log₁₀(y)

• Natural Logarithm: The natural logarithm (ln) has a base of e (Euler's number, approximately 2.71828). ln(y) = log_e(y)

• Properties of Logarithms:

  • Product Rule: log_b(mn) = log_b(m) + log_b(n)
  • Quotient Rule: log_b(m/n) = log_b(m) - log_b(n)
  • Power Rule: log_b(m^p) = plog_b(m)*
  • Change of Base Formula: log_a(b) = log_c(b) / log_c(a)

Advanced Exponential Equations

Some exponential equations require more advanced techniques, such as substitution or factoring.

• Example: Solve e^2x - 3e^x + 2 = 0

  • Let y = e^x. Then the equation becomes: y² - 3y + 2 = 0
  • Factor the quadratic: (y - 1)(y - 2) = 0
  • Solve for y: y = 1 or y = 2
  • Substitute back e^x for y: e^x = 1 or e^x = 2
  • Solve for x: x = ln(1) = 0 or x = ln(2) ≈ 0.693

Homework Problem-Solving Strategies

Now that you have a strong foundation in exponents and exponential functions, let's talk about strategies for tackling your homework:

1. Read the instructions carefully: Make sure you understand what the problem is asking you to do.
2. Identify the key concepts: Determine which exponent rules or properties of exponential functions are relevant to the problem.
3. Simplify the expression: Use exponent rules to simplify expressions as much as possible before attempting to solve for the variable.
4. Isolate the exponential term: If you're solving an exponential equation, try to isolate the exponential term on one side of the equation.
5. Choose the appropriate method: Decide whether you can equate the bases or if you need to use logarithms to solve for the variable.
6. Check your answer: Plug your solution back into the original equation to make sure it's correct.
7. Show your work: Even if you can solve a problem in your head, it's important to show your work so that your teacher can see your reasoning.

Common Mistakes to Avoid

• Incorrectly applying exponent rules: Double-check that you are using the correct rule for each situation. For example, b^m * bⁿ = b^m+n, not b^mn*.
• Forgetting the order of operations: Remember to follow the order of operations (PEMDAS/BODMAS) when simplifying expressions.
• Making sign errors: Be careful when dealing with negative exponents and negative numbers.
• Incorrectly using logarithms: Make sure you understand the properties of logarithms and how to apply them correctly.
• Not checking your answer: Always plug your solution back into the original equation to make sure it's correct.
• Rounding errors: If you need to round your answer, do it at the very end of the problem to minimize rounding errors.

Let's Summarize: Key Takeaways

• Exponents are shorthand for repeated multiplication. Mastering exponent rules is crucial.
• Exponential functions model growth and decay. The base b determines whether the function is growing (b > 1) or decaying (0 < b < 1).
• Logarithms are the inverse of exponential functions. They are essential for solving exponential equations where you can't easily equate the bases.
• Practice is key! The more you practice solving problems, the better you will become at understanding and applying these concepts.

By understanding the fundamental principles, practicing regularly, and avoiding common mistakes, you'll be well-equipped to ace your Unit 6 homework and gain a solid understanding of exponents and exponential functions! Good luck!