Integer Exponents

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Answer Key & Explanations

Integer Exponents · Grade 8 · Worksheet 2

1. A scientist is studying bacterial growth in a lab. The bacteria population starts with 2,000 cells and doubles every hour. After 5 hours, the scientist needs to express the final population in scientific notation. What is the population of bacteria after 5 hours, written in proper scientific notation? Answer: 6.4 × 10^4 Solution: We start with 2,000 bacteria. The population doubles every hour. We want the population after 5 hours.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We start with 2,000 bacteria. The population doubles every hour. We want the population after 5 hours. --- **Step 2: Write the exponential growth formula** Initial population = \( P_0 = 2000 \) Doubling time = 1 hour Time elapsed = \( t = 5 \) hours Population after \( t \) hours: \[ P(t) = P_0 \times 2^{t} \] Substitute: \[ P(5) = 2000 \times 2^{5} \] --- **Step 3: Calculate \( 2^5 \)** \[ 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \] So: \[ P(5) = 2000 \times 32 \] --- **Step 4: Multiply 2000 by 32** \[ 2000 \times 32 = 2000 \times (30 + 2) = 2000 \times 30 + 2000 \times 2 = 60000 + 4000 = 64000 \] So the population is 64,000. --- **Step 5: Convert 64,000 to scientific notation** Scientific notation: \( a \times 10^n \) where \( 1 \leq a < 10 \). 64,000 = 64 × 1000 = 64 × \( 10^3 \) But 64 = 6.4 × 10, so: 64 × \( 10^3 \) = (6.4 × 10) × \( 10^3 \) = 6.4 × \( 10^{1+3} \) = 6.4 × \( 10^4 \). Check: 6.4 × 10000 = 64000. Correct. --- **Step 6: Final answer** \[ 6.4 \times 10^4 \]

2. A scientist is studying bacterial growth in a lab. The bacteria population starts with 2^3 cells and doubles every hour according to the pattern 2^(3+n), where n is the number of hours. After 4 hours, the scientist adds a nutrient that multiplies the current population by 2^5 cells. What is the total number of bacteria cells after this nutrient is added? Express your answer in exponential form with a single base. Answer: 2^12 Solution: The formula given is \( 2^{3+n} \), where \( n \) is the number of hours. At \( n = 0 \) (start): \( 2^{3+0} = 2^3 \) — matches the initial count.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the initial population and growth pattern** The bacteria start with \( 2^3 \) cells. The formula given is \( 2^{3+n} \), where \( n \) is the number of hours. At \( n = 0 \) (start): \( 2^{3+0} = 2^3 \) — matches the initial count. --- **Step 2: Find the population after 4 hours** After \( n = 4 \) hours: Population = \( 2^{3+4} = 2^7 \). So after 4 hours, there are \( 2^7 \) cells. --- **Step 3: Apply the nutrient effect** The nutrient multiplies the current population by \( 2^5 \) cells. "Multiplies by \( 2^5 \) cells" means multiply by \( 2^5 \). So: New population = \( 2^7 \times 2^5 \). --- **Step 4: Combine exponents** Using the rule \( a^m \times a^n = a^{m+n} \): \( 2^7 \times 2^5 = 2^{7+5} = 2^{12} \). --- **Step 5: Final answer** The total number of bacteria cells after the nutrient is added is \( 2^{12} \). --- **Answer:** 2^12

3. A rectangular solar panel has dimensions of 8.4 × 10^3 millimeters by 3.6 × 10^2 millimeters. What is the area of the solar panel in square millimeters, expressed in standard scientific notation? Answer: 3.024×10^6 Solution: Step 1: Write the area formula: Area = length × width Step 2: Substitute the given values: Area = (8.4 × 10^3) × (3.6 × 10^2) Step 3: Multiply the coefficients: 8.4 × 3.6 = 30.24 Step 4: Add the exponents: 10^3 × 10^2 = 10^(3+2) = 10^5 Step 5: Combine the results: 30.24 × 10^5 Step 6: Convert to…
   Full step-by-step solution

   Step 1: Write the area formula: Area = length × width Step 2: Substitute the given values: Area = (8.4 × 10^3) × (3.6 × 10^2) Step 3: Multiply the coefficients: 8.4 × 3.6 = 30.24 Step 4: Add the exponents: 10^3 × 10^2 = 10^(3+2) = 10^5 Step 5: Combine the results: 30.24 × 10^5 Step 6: Convert to standard scientific notation: 30.24 × 10^5 = 3.024 × 10^6 Step 7: Verify that the coefficient is between 1 and 10: 3.024 is between 1 and 10 The final answer is 3.024×10^6.

4. A scientist is studying bacterial growth in a lab. The initial population is 200 bacteria, and it doubles every 3 hours. The scientist uses the formula P = 200 × 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: 3200 Solution: P = 200 × 2^(t/3) - P = population after t hours - 200 = initial population - t = time in hours - The doubling time is 3 hours, so t/3 tells how many doubling periods have passed.
   Full step-by-step solution

   Let's go step by step. --- **Step 1: Understand the formula** The formula given is: P = 200 × 2^(t/3) Where: - P = population after t hours - 200 = initial population - t = time in hours - The doubling time is 3 hours, so t/3 tells how many doubling periods have passed. --- **Step 2: Identify the given value** t = 12 hours --- **Step 3: Substitute t into the formula** P = 200 × 2^(12/3) --- **Step 4: Simplify the exponent** 12/3 = 4 So: P = 200 × 2^4 --- **Step 5: Calculate 2^4** 2^4 = 2 × 2 × 2 × 2 = 16 --- **Step 6: Multiply** P = 200 × 16 200 × 16 = 3200 --- **Step 7: Final answer** The total population after 12 hours is 3200 bacteria.

5. (3⁵ × 3⁻³) ÷ (3⁴ × 3⁻²) = ? Answer: 1 Solution: Simplify the numerator using x^m · x^n = x^(m+n): 3⁵ × 3⁻³ = 3^(5 + (-3)) = 3² Simplify the denominator: 3⁴ × 3⁻² = 3^(4 + (-2)) = 3² Divide: 3² ÷ 3² = 3^(2-2) = 3⁰ Any nonzero number to the power of 0 equals 1: 3⁰ = 1 The answer is 1.
   Full step-by-step solution

   Step 1: Simplify the numerator using x^m · x^n = x^(m+n): 3⁵ × 3⁻³ = 3^(5 + (-3)) = 3² Step 2: Simplify the denominator: 3⁴ × 3⁻² = 3^(4 + (-2)) = 3² Step 3: Divide: 3² ÷ 3² = 3^(2-2) = 3⁰ Step 4: Any nonzero number to the power of 0 equals 1: 3⁰ = 1 The answer is 1.

6. (2³ × 4²) ÷ (2² × 2³) = ? Answer: 4 Solution: Write all terms with base 2. Note that 4² = (2²)² = 2⁴.
   Full step-by-step solution

   Step 1: Write all terms with base 2. Note that 4² = (2²)² = 2⁴. Step 2: Rewrite the expression: (2³ × 2⁴) ÷ (2² × 2³) Step 3: Simplify numerator: 2³ × 2⁴ = 2^(3+4) = 2⁷ Step 4: Simplify denominator: 2² × 2³ = 2^(2+3) = 2⁵ Step 5: Divide: 2⁷ ÷ 2⁵ = 2^(7-5) = 2² Step 6: Calculate 2² = 4 The answer is 4.

7. Liam is studying the growth of algae in a pond for his science project. He observes that the algae population starts at 5,000 cells and quadruples every 2 days. After 6 days, a natural predator reduces the population to one-sixteenth of its current size. What is the final algae population after this reduction? Express your answer in standard form. Answer: 20000 Solution: Determine the growth pattern. The algae quadruples every 2 days, so the growth factor per 2-day period is 4. Calculate the population after 6 days.
   Full step-by-step solution

   Step 1: Determine the growth pattern. The algae quadruples every 2 days, so the growth factor per 2-day period is 4. Step 2: Calculate the population after 6 days. Since 6 days represents 3 periods of 2 days each, the population grows by a factor of 4^3. Step 3: Initial population × growth factor = 5,000 × 4^3 = 5,000 × 64 = 320,000 cells after 6 days. Step 4: Apply the reduction. The predator reduces the population to one-sixteenth, so multiply by 1/16. Step 5: Final population = 320,000 × (1/16) = 320,000 ÷ 16 = 20,000 cells. The final algae population is 20,000 cells.