Integer Exponents

lessonbunny.com

Answer Key & Explanations

Integer Exponents · Grade 8 · Worksheet 1

1. Liam is studying the growth of algae in a pond. He starts with an initial population of 1.6 × 10^4 algae cells. The population triples every day. After 3 days, a chemical treatment is applied that reduces the population to one-twenty-seventh of its current size. What is the final population of algae cells? Express your answer in proper scientific notation. Answer: 1.6 × 10^4 Solution: Start with initial population: 1.6 × 10^4 Population triples daily for 3 days: (1.6 × 10^4) × 3^3 = (1.6 × 10^4) × 27 After 3 days, population is reduced to 1/27 of its size: [(1.6 × 10^4) × 27] × (1/27) Simplify using properties of exponents: 27 × (1/27) = 1 Final population = 1.6 × 10^4 × 1 =…
   Full step-by-step solution

   Step 1: Start with initial population: 1.6 × 10^4 Step 2: Population triples daily for 3 days: (1.6 × 10^4) × 3^3 = (1.6 × 10^4) × 27 Step 3: After 3 days, population is reduced to 1/27 of its size: [(1.6 × 10^4) × 27] × (1/27) Step 4: Simplify using properties of exponents: 27 × (1/27) = 1 Step 5: Final population = 1.6 × 10^4 × 1 = 1.6 × 10^4 The answer is 1.6 × 10^4.

2. (5³ × 5⁻²) ÷ (5² × 5⁻⁴) = ? Answer: 125 Solution: Step 1: Apply the product of powers property to the numerator: 5³ × 5⁻² = 5^(3 + (-2)) = 5¹ Step 2: Apply the product of powers property to the denominator: 5² × 5⁻⁴ = 5^(2 + (-4)) = 5⁻² Step 3: Now we have the division: 5¹ ÷ 5⁻² Step 4: Apply the quotient of powers property: 5^(1 - (-2)) = 5^(1…
   Full step-by-step solution

   Step 1: Apply the product of powers property to the numerator: 5³ × 5⁻² = 5^(3 + (-2)) = 5¹ Step 2: Apply the product of powers property to the denominator: 5² × 5⁻⁴ = 5^(2 + (-4)) = 5⁻² Step 3: Now we have the division: 5¹ ÷ 5⁻² Step 4: Apply the quotient of powers property: 5^(1 - (-2)) = 5^(1 + 2) = 5³ Step 5: Calculate 5³ = 5 × 5 × 5 = 25 × 5 = 125 The answer is 125.

3. A scientist is studying bacterial growth in a lab. The initial population is 500 bacteria, and it doubles every 3 hours. The scientist uses the formula P = 500 × 2^(t/3) to model the population after t hours. If the experiment runs for 12 hours, what will be the total population of bacteria? Express your answer as a whole number. Answer: 8000 Solution: Initial population = 500 Doubling time = 3 hours Formula: P = 500 × 2^(t/3) Time t = 12 hours Write down the formula with the given values. P = 500 × 2^(12/3) Simplify the exponent.
   Full step-by-step solution

   Let's go step-by-step. We are given: Initial population = 500 Doubling time = 3 hours Formula: P = 500 × 2^(t/3) Time t = 12 hours --- **Step 1: Write down the formula with the given values.** P = 500 × 2^(12/3) --- **Step 2: Simplify the exponent.** 12/3 = 4 So P = 500 × 2^4 --- **Step 3: Calculate 2^4.** 2^4 = 2 × 2 × 2 × 2 = 16 --- **Step 4: Multiply 500 by 16.** 500 × 16 = 8000 --- **Step 5: Interpret the result.** After 12 hours, the population is 8000 bacteria. --- **Final answer:** 8000

4. (3⁵ × 3⁻³) ÷ (3² × 3⁻⁴) = ? Answer: 81 Solution: Simplify the numerator: 3⁵ × 3⁻³ = 3^(5 + (-3)) = 3² Simplify the denominator: 3² × 3⁻⁴ = 3^(2 + (-4)) = 3⁻² Now we have 3² ÷ 3⁻² = 3^(2 - (-2)) = 3^(2 + 2) = 3⁴ Calculate 3⁴ = 3 × 3 × 3 × 3 = 9 × 9 = 81 The answer is 81.
   Full step-by-step solution

   Step 1: Simplify the numerator: 3⁵ × 3⁻³ = 3^(5 + (-3)) = 3² Step 2: Simplify the denominator: 3² × 3⁻⁴ = 3^(2 + (-4)) = 3⁻² Step 3: Now we have 3² ÷ 3⁻² = 3^(2 - (-2)) = 3^(2 + 2) = 3⁴ Step 4: Calculate 3⁴ = 3 × 3 × 3 × 3 = 9 × 9 = 81 The answer is 81.

5. (2⁷ × 2⁻²) ÷ (2³ × 2⁻⁶) = ? Answer: 256 Solution: Simplify the numerator: 2⁷ × 2⁻² = 2^(7 + (-2)) = 2⁵ Simplify the denominator: 2³ × 2⁻⁶ = 2^(3 + (-6)) = 2⁻³ Now we have 2⁵ ÷ 2⁻³ = 2^(5 - (-3)) = 2^(5 + 3) = 2⁸ Calculate 2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256 The answer is 256.
   Full step-by-step solution

   Step 1: Simplify the numerator: 2⁷ × 2⁻² = 2^(7 + (-2)) = 2⁵ Step 2: Simplify the denominator: 2³ × 2⁻⁶ = 2^(3 + (-6)) = 2⁻³ Step 3: Now we have 2⁵ ÷ 2⁻³ = 2^(5 - (-3)) = 2^(5 + 3) = 2⁸ Step 4: Calculate 2⁸ = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 256 The answer is 256.

6. A rectangular prism has dimensions of 3 × 10⁴ cm by 2 × 10² cm by 5 × 10³ cm. Using the properties of exponents, calculate the volume of the prism in scientific notation. Answer: 3 × 10¹⁰ Solution: Write the volume formula: Volume = length × width × height Substitute the given dimensions: Volume = (3 × 10⁴) × (2 × 10²) × (5 × 10³) Multiply the coefficients: 3 × 2 × 5 = 30 Multiply the powers of 10 using the exponent rule: 10⁴ × 10² × 10³ = 10^(4+2+3) = 10⁹ Combine the results: 30 × 10⁹…
   Full step-by-step solution

   Step 1: Write the volume formula: Volume = length × width × height Step 2: Substitute the given dimensions: Volume = (3 × 10⁴) × (2 × 10²) × (5 × 10³) Step 3: Multiply the coefficients: 3 × 2 × 5 = 30 Step 4: Multiply the powers of 10 using the exponent rule: 10⁴ × 10² × 10³ = 10^(4+2+3) = 10⁹ Step 5: Combine the results: 30 × 10⁹ Step 6: Convert to proper scientific notation: 3.0 × 10¹ × 10⁹ = 3.0 × 10^(1+9) = 3.0 × 10¹⁰ The answer is 3 × 10¹⁰.

7. Hana is designing a rectangular garden for her school project. The length of the garden is 4^3 meters, and the width is 4^2 meters. She wants to cover the entire garden with grass seeds. Each packet of seeds covers an area of 4^4 square meters. How many packets of seeds does Hana need to buy? Express your answer in exponential form with a base of 4. Answer: 4^1 Solution: Find the area of the garden. Area = length × width = 4^3 × 4^2. Using the product of powers property (x^m · x^n = x^(m+n)), we get 4^(3+2) = 4^5 square meters.
   Full step-by-step solution

   Step 1: Find the area of the garden. Area = length × width = 4^3 × 4^2. Using the product of powers property (x^m · x^n = x^(m+n)), we get 4^(3+2) = 4^5 square meters. Step 2: Each seed packet covers 4^4 square meters. To find the number of packets needed, divide the total area by the area per packet: 4^5 ÷ 4^4. Step 3: Using the quotient of powers property (x^m ÷ x^n = x^(m-n)), we get 4^(5-4) = 4^1. Step 4: So Hana needs 4^1 packet of seeds. Answer: 4^1

8. The population of a certain bacteria colony doubles every hour. The initial population is 5 × 10³ bacteria. After 4 hours, the population is given by the expression (5 × 10³) × 2⁴. Simplify this expression and write the final population in standard scientific notation (a × 10^b where 1 ≤ a < 10). Answer: 8e7 Solution: The solution involves simplifying the product of a number in scientific notation and a power, then converting the result back into standard scientific notation.
   Full step-by-step solution

   This problem applies the properties of exponents, specifically the product of powers and power of a power rules, to a real-world scenario of exponential growth. The solution involves simplifying the product of a number in scientific notation and a power, then converting the result back into standard scientific notation.