Unit 6 Exponents And Exponential Functions Homework 1 Answer Key

Unit 6: Exponents and Exponential Functions - Homework 1: A Comprehensive Guide

This guide provides detailed solutions and explanations for a typical Unit 6, Homework 1 assignment covering exponents and exponential functions. Remember that specific problems will vary depending on your textbook and curriculum, but the concepts and methods explained here are broadly applicable. We'll cover fundamental exponent rules, simplifying expressions, solving exponential equations, and graphing exponential functions. Use this as a resource to check your work, understand the underlying concepts, and master this crucial area of algebra.

Section 1: Review of Exponent Rules

Before tackling the homework problems, let's solidify our understanding of the fundamental rules governing exponents. These rules are the building blocks for solving more complex problems.

1. Product Rule: When multiplying terms with the same base, add the exponents: a<sup>m</sup> * a<sup>n</sup> = a<sup>m+n</sup>

Example: x<sup>3</sup> * x<sup>5</sup> = x<sup>3+5</sup> = x<sup>8</sup>

2. Quotient Rule: When dividing terms with the same base, subtract the exponents: a<sup>m</sup> / a<sup>n</sup> = a<sup>m-n</sup>

Example: y<sup>7</sup> / y<sup>2</sup> = y<sup>7-2</sup> = y<sup>5</sup>

3. Power Rule: When raising a power to another power, multiply the exponents: (a<sup>m</sup>)<sup>n</sup> = a<sup>m*n</sup>

Example: (z<sup>4</sup>)<sup>3</sup> = z<sup>4*3</sup> = z<sup>12</sup>

4. Zero Exponent Rule: Any non-zero base raised to the power of zero equals 1: a<sup>0</sup> = 1 (where a ≠ 0)

Example: 5<sup>0</sup> = 1

5. Negative Exponent Rule: A negative exponent indicates a reciprocal: a<sup>-n</sup> = 1/a<sup>n</sup>

Example: x<sup>-2</sup> = 1/x<sup>2</sup>

6. Power of a Product Rule: When raising a product to a power, raise each factor to that power: (ab)<sup>n</sup> = a<sup>n</sup>b<sup>n</sup>

Example: (2x)<sup>3</sup> = 2<sup>3</sup>x<sup>3</sup> = 8x<sup>3</sup>

7. Power of a Quotient Rule: When raising a quotient to a power, raise both the numerator and the denominator to that power: (a/b)<sup>n</sup> = a<sup>n</sup>/b<sup>n</sup> (where b ≠ 0)

Example: (x/y)<sup>4</sup> = x<sup>4</sup>/y<sup>4</sup>

Section 2: Simplifying Exponential Expressions

Homework problems often involve simplifying expressions using the rules above. Let's work through a few examples:

Problem 1: Simplify (2x<sup>2</sup>y<sup>3</sup>)<sup>2</sup> * (3xy<sup>4</sup>)<sup>3</sup>

Solution:

1. Apply the Power of a Product Rule: (2²x⁴y⁶) * (3³x³y¹²)
2. Simplify the numerical coefficients: 4x⁴y⁶ * 27x³y¹²
3. Apply the Product Rule for the x terms and y terms: 108x⁷y¹⁸

Therefore, the simplified expression is 108x<sup>7</sup>y<sup>18</sup>

Problem 2: Simplify (a<sup>3</sup>b<sup>-2</sup>c<sup>4</sup>) / (a<sup>-1</sup>b<sup>2</sup>c<sup>-1</sup>)

Solution:

1. Apply the Quotient Rule for each variable: a^3-(-1)b^-2-2c^4-(-1)
2. Simplify the exponents: a⁴b^-4c⁵
3. Rewrite with positive exponents: a⁴c⁵/b⁴

Therefore, the simplified expression is a<sup>4</sup>c<sup>5</sup>/b<sup>4</sup>

Section 3: Solving Exponential Equations

Exponential equations involve variables in the exponent. Solving them often requires using the properties of exponents and sometimes logarithms (which you might cover later in the unit).

Problem 3: Solve for x: 2<sup>x</sup> = 16

Solution:

Rewrite 16 as a power of 2: 16 = 2⁴

Therefore, 2<sup>x</sup> = 2<sup>4</sup>

Since the bases are equal, the exponents must be equal: x = 4

Problem 4: Solve for x: 3<sup>x+1</sup> = 27

Solution:

Rewrite 27 as a power of 3: 27 = 3³

Therefore, 3<sup>x+1</sup> = 3<sup>3</sup>

Equate the exponents: x + 1 = 3

Solve for x: x = 2

Problem 5: (More advanced, potentially requiring logarithms): Solve for x: 5<sup>x</sup> = 12

Solution: This problem requires the use of logarithms. You would take the logarithm (base 10 or natural log) of both sides:

log(5^x) = log(12)

Using logarithm properties: x * log(5) = log(12)

Solve for x: x = log(12) / log(5) (Use a calculator to find the numerical value)

Section 4: Graphing Exponential Functions

Exponential functions have the general form f(x) = a * b<sup>x</sup>, where 'a' is the initial value and 'b' is the base. The graph's behavior depends on the value of 'b'.

• If b > 1: The graph shows exponential growth (increasing).
• If 0 < b < 1: The graph shows exponential decay (decreasing).

Problem 6: Graph the function f(x) = 2<sup>x</sup>

Solution:

Create a table of values:

Plot these points on a graph. The graph will show exponential growth, starting from close to zero for negative x values and increasing rapidly for positive x values. The graph will never touch the x-axis (asymptote).

Problem 7: Graph the function f(x) = (1/2)<sup>x</sup>

Solution:

Create a table of values:

Plot these points. The graph shows exponential decay; it starts from a high value for negative x and decreases rapidly towards zero as x increases. The graph will never touch the x-axis (asymptote).

Section 5: Word Problems involving Exponents and Exponential Functions

Many real-world scenarios can be modeled using exponential functions, such as compound interest, population growth, or radioactive decay.

Problem 8: A bacteria population doubles every hour. If there are initially 100 bacteria, how many will there be after 5 hours?

Solution:

This is an exponential growth problem. The formula is: Population = Initial Population * 2^{number of hours}

Population = 100 * 2⁵ = 100 * 32 = 3200 bacteria

Problem 9: The value of a car depreciates by 15% each year. If it was initially worth $20,000, what will its value be after 3 years?

Solution:

This is an exponential decay problem. The remaining value after each year is 85% (100% - 15%). The formula is: Value = Initial Value * (0.85)^{number of years}

Value = 20000 * (0.85)³ = 20000 * 0.614125 ≈ $12,282.50

This comprehensive guide provides a strong foundation for understanding and solving problems related to exponents and exponential functions. Remember to practice regularly, review the exponent rules, and don't hesitate to seek clarification if needed. By mastering these concepts, you’ll be well-equipped to tackle more advanced topics in algebra and beyond. Good luck with your homework!