Unit 6 Exponents And Exponential Functions Homework 9 Answer Key

Embark on a journey into the realm of exponents and exponential functions, where the principles of mathematics unlock patterns of growth and decay that shape our world. This exploration delves into the intricacies of Unit 6, Homework 9, offering an answer key that illuminates the underlying concepts and provides clarity on the applications of exponential functions.

Understanding Exponents

Exponents are a shorthand notation for repeated multiplication. The expression aⁿ, where a is the base and n is the exponent, signifies that a is multiplied by itself n times. For instance, 2³ = 2 × 2 × 2 = 8.

The Laws of Exponents

These laws govern how exponents behave in various mathematical operations:

• Product of Powers: a^m × aⁿ = a^{m + n} (When multiplying powers with the same base, add the exponents)
• Quotient of Powers: a^m / aⁿ = a^{m - n} (When dividing powers with the same base, subtract the exponents)
• Power of a Power: (a^m)ⁿ = a^{m × n} (When raising a power to another power, multiply the exponents)
• Power of a Product: (ab)ⁿ = aⁿbⁿ (The power of a product is the product of the powers)
• Power of a Quotient: (a/ b)ⁿ = aⁿ/ bⁿ (The power of a quotient is the quotient of the powers)
• Zero Exponent: a⁰ = 1 (Any nonzero number raised to the power of 0 equals 1)
• Negative Exponent: a^-n = 1/ aⁿ (A negative exponent indicates the reciprocal of the base raised to the positive exponent)
• Fractional Exponent: a^{m/ n} = ⁿ√(a^m) (A fractional exponent represents a radical)

Exponential Functions: A Definition

An exponential function is a mathematical function in which the independent variable (x) appears in the exponent. A general form of an exponential function is:

f(x) = a^x

where a is a constant called the base, and x is the exponent. The base a must be positive and not equal to 1.

Key Properties of Exponential Functions

• Domain: The domain of an exponential function is all real numbers.
• Range: If a > 0, the range is all positive real numbers. If a < 0, the range is all negative real numbers.
• Horizontal Asymptote: Exponential functions have a horizontal asymptote at y = 0.
• Intercept: The y-intercept is always (0, 1) because a⁰ = 1.
• Growth/Decay: If a > 1, the function represents exponential growth. If 0 < a < 1, the function represents exponential decay.

Unit 6 Homework 9: Answer Key and Explanations

This section provides solutions to typical problems found in Unit 6 Homework 9, covering various aspects of exponents and exponential functions.

Problem 1: Simplify the expression (3x²y^-1)³

Solution:

Applying the power of a product rule, we get:

(3x²y^-1)³ = 3³ * (x²)³ * (y^-1)³

= 27 * x^2×3 * y^-1×3

= 27x⁶y^-3

Using the negative exponent rule, we rewrite y^-3 as 1/y³:

= 27x⁶/ y³

Therefore, the simplified expression is 27x⁶/ y³.

Problem 2: Evaluate the expression 16^3/4

Solution:

We can rewrite 16^3/4 as (16^1/4)³.

16^1/4 is the fourth root of 16, which is 2, because 2⁴ = 16.

Therefore, (16^1/4)³ = 2³ = 8.

The evaluated expression is 8.

Problem 3: Solve for x in the equation 2^x = 32

Solution:

We need to express 32 as a power of 2. We know that 32 = 2⁵.

Therefore, the equation becomes 2^x = 2⁵.

Since the bases are equal, the exponents must be equal:

x = 5

The solution is x = 5.

Problem 4: Solve for x in the equation 9^x = 1/27

Solution:

Express both 9 and 27 as powers of 3. We have 9 = 3² and 27 = 3³. Therefore, 1/27 = 3^-3.

The equation becomes (3²)^x = 3^-3, which simplifies to 3^2x = 3^-3.

Since the bases are equal, the exponents must be equal:

2x = -3

Divide both sides by 2:

x = -3/2

The solution is x = -3/2.

Problem 5: Sketch the graph of the exponential function f(x) = 2^x - 3. Identify the horizontal asymptote and y-intercept.

Solution:

• Horizontal Asymptote: The horizontal asymptote of f(x) = 2^x is y = 0. However, due to the vertical shift of -3, the horizontal asymptote is y = -3.

• Y-intercept: To find the y-intercept, set x = 0:

  f(0) = 2⁰ - 3 = 1 - 3 = -2

  Therefore, the y-intercept is (0, -2).

The graph of f(x) = 2^x - 3 is an exponential growth curve shifted downward by 3 units. It passes through (0, -2) and approaches the horizontal asymptote y = -3 as x approaches negative infinity.

Problem 6: A population of bacteria doubles every 3 hours. If the initial population is 100, what is the population after 12 hours?

Solution:

The exponential growth can be modeled by the equation P(t) = P₀ * 2^{t/ d}, where P(t) is the population at time t, P₀ is the initial population, and d is the doubling time.

In this case, P₀ = 100, and d = 3 hours. We want to find the population after t = 12 hours.

P(12) = 100 * 2^12/3

= 100 * 2⁴

= 100 * 16

= 1600

Therefore, the population after 12 hours is 1600 bacteria.

Problem 7: The half-life of a radioactive substance is 50 years. If you start with 200 grams of the substance, how much will remain after 150 years?

Solution:

The exponential decay can be modeled by the equation A(t) = A₀ * (1/2)^{t/ h}, where A(t) is the amount remaining at time t, A₀ is the initial amount, and h is the half-life.

In this case, A₀ = 200 grams, and h = 50 years. We want to find the amount remaining after t = 150 years.

A(150) = 200 * (1/2)^150/50

= 200 * (1/2)³

= 200 * (1/8)

= 25

Therefore, 25 grams of the substance will remain after 150 years.

Problem 8: Write an exponential function to represent the following data:

x	y
0	3
1	6
2	12
3	24

Solution:

The general form of an exponential function is f(x) = a * b^x.

When x = 0, f(0) = 3, so a * b⁰ = 3, which means a * 1 = 3, and therefore a = 3.

Now we need to find b. Using the point (1, 6), we have f(1) = 6, so 3 * b¹ = 6, which means 3b = 6, and therefore b = 2.

Thus, the exponential function that represents the data is f(x) = 3 * 2^x.

Problem 9: Simplify the expression (4x^-2y³z⁰) / (2x⁴y^-1)

Solution:

First, simplify the constants: 4/2 = 2.

Then, apply the quotient of powers rule for each variable:

• x^-2 / x⁴ = x^-2-4 = x^-6
• y³ / y^-1 = y^3-(-1) = y⁴
• z⁰ = 1 (any non-zero number raised to the power of 0 is 1)

Putting it all together:

(4x^-2y³z⁰) / (2x⁴y^-1) = 2 * x^-6 * y⁴ * 1

= 2y⁴/ x⁶

Therefore, the simplified expression is 2y⁴/ x⁶.

Problem 10: Solve for x: 5^x+1 = 25^x-2

Solution:

Express both sides with the same base. Since 25 = 5², we can rewrite the equation as:

5^x+1 = (5²)^x-2

Using the power of a power rule, we get:

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

5^x+1 = 5^2x-4

Since the bases are equal, the exponents must be equal:

x + 1 = 2x - 4

Subtract x from both sides:

1 = x - 4

Add 4 to both sides:

5 = x

Therefore, the solution is x = 5.

Practical Applications of Exponential Functions

Exponential functions are not just theoretical constructs; they are powerful tools used to model real-world phenomena:

• Population Growth: As seen in the bacteria example, exponential functions can describe how populations grow over time, assuming unlimited resources.
• Radioactive Decay: The decay of radioactive isotopes follows an exponential decay model, used in carbon dating and medical imaging.
• Compound Interest: The growth of money in a bank account with compound interest is an exponential function. The more frequently interest is compounded, the faster the money grows.
• Spread of Diseases: The spread of infectious diseases can sometimes be modeled using exponential functions, especially in the early stages of an outbreak.
• Learning Curves: In psychology and education, exponential functions can describe how quickly someone learns a new skill or concept. The initial learning is often rapid, but the rate of learning slows down over time.
• Cooling and Heating: Newton's Law of Cooling describes how the temperature of an object changes over time as it approaches the temperature of its surroundings. This is modeled using an exponential function.
• Financial Modeling: Predicting the growth of investments, calculating mortgage payments, and analyzing financial risk all involve exponential functions.

Common Mistakes and How to Avoid Them

Understanding the laws of exponents and the properties of exponential functions is critical for avoiding common mistakes:

• Incorrectly Applying the Laws of Exponents: Ensure you apply the correct law for each operation. For example, don't confuse the product of powers rule with the power of a power rule.
• Forgetting the Zero Exponent Rule: Any non-zero number raised to the power of 0 equals 1. This is a fundamental rule.
• Misinterpreting Negative Exponents: Remember that a negative exponent means taking the reciprocal of the base raised to the positive exponent.
• Ignoring Order of Operations: Follow the correct order of operations (PEMDAS/BODMAS) when simplifying expressions involving exponents.
• Confusing Exponential and Linear Functions: Exponential functions involve a constant base raised to a variable exponent, while linear functions have a constant rate of change.
• Incorrectly Identifying Growth vs. Decay: In exponential functions, a base greater than 1 indicates growth, while a base between 0 and 1 indicates decay.

Advanced Topics: Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. They are essential for solving exponential equations where the variable is in the exponent and cannot be easily isolated.

The logarithmic function y = log_a(x) is equivalent to the exponential function x = a^y. In other words, the logarithm of x to the base a is the exponent to which a must be raised to produce x.

Properties of Logarithms

Similar to exponents, logarithms have their own set of properties:

• Product Rule: log_a(xy) = log_a(x) + log_a(y)
• Quotient Rule: log_a(x/ y) = log_a(x) - log_a(y)
• Power Rule: log_a(xⁿ) = n log_a(x)
• Change of Base Rule: log_b(x) = log_a(x) / log_a(b)

Solving Exponential Equations Using Logarithms

Logarithms are crucial for solving exponential equations where isolating the variable in the exponent is not straightforward. For example, consider the equation 3^x = 10.

To solve for x, take the logarithm of both sides (using any base, but base 10 or base e (natural logarithm) are common):

log(3^x) = log(10)

Using the power rule of logarithms:

x log(3) = log(10)

Divide both sides by log(3):

x = log(10) / log(3)

Using a calculator, we find that x ≈ 2.096.

Conclusion

Mastering exponents and exponential functions opens doors to understanding a wide array of phenomena in mathematics, science, and finance. By grasping the fundamental laws of exponents, recognizing the properties of exponential functions, and practicing problem-solving techniques, you can confidently tackle Unit 6 Homework 9 and beyond. Remember to pay attention to detail, avoid common mistakes, and explore the connections between exponential and logarithmic functions for a deeper understanding of these powerful mathematical tools.