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Integer Exponents

Grade 8 · Algebra · Worksheet 3

  1. A rectangular prism has dimensions of 6 × 10² meters by 3 × 10³ meters by 2 × 10¹ meters. Using the properties of exponents, calculate the volume of the prism in scientific notation. Answer: ______________
  2. (6⁴ × 6⁻²) ÷ (6³ × 6⁻⁴) = ? Answer: ______________
  3. (5³ × 5⁻²) ÷ (5² × 5⁻⁴) = ? Answer: ______________
  4. Liam is studying the growth of algae in a pond for his science project. He observes that the algae population starts at 8,000 cells and triples every 2 hours. After 6 hours, a special treatment is applied that reduces the population to one-ninth of its size at that moment. What is the final population of algae cells? Express your answer in standard form. Answer: ______________
  5. Sophia is designing a rectangular garden for her school's science project. The length of the garden is 4^3 feet and the width is 4^1 feet. She plans to cover the entire garden with soil. What is the area of the garden in square feet? Express your answer as a single power of 4. Answer: ______________
  6. Matiu is helping his school's technology club design a sound amplifier. The amplifier's power output in watts can be modeled by the expression (3^4 × 3^2)^3. Matiu needs to simplify this expression to find the power output in exponential form with a single base. What is the simplified expression? Answer: ______________
  7. (9³ × 9⁻⁴) ÷ (9⁻² × 9⁰) = ? Answer: ______________
  8. (6³ × 6⁻¹) ÷ (6² × 6⁻⁴) = ? Answer: ______________
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Answer Key & Explanations

Integer Exponents · Grade 8 · Worksheet 3

  1. A rectangular prism has dimensions of 6 × 10² meters by 3 × 10³ meters by 2 × 10¹ meters. Using the properties of exponents, calculate the volume of the prism in scientific notation. Answer: 3.6 × 10⁷ Solution: Write the volume formula: Volume = length × width × height Substitute the given dimensions: Volume = (6 × 10²) × (3 × 10³) × (2 × 10¹) Multiply the coefficients: 6 × 3 × 2 = 36 Multiply the powers of 10: 10² × 10³ × 10¹ = 10^(2+3+1) = 10⁶ Combine results: 36 × 10⁶ Convert to proper scientific…
    Full step-by-step solution

    Step 1: Write the volume formula: Volume = length × width × height Step 2: Substitute the given dimensions: Volume = (6 × 10²) × (3 × 10³) × (2 × 10¹) Step 3: Multiply the coefficients: 6 × 3 × 2 = 36 Step 4: Multiply the powers of 10: 10² × 10³ × 10¹ = 10^(2+3+1) = 10⁶ Step 5: Combine results: 36 × 10⁶ Step 6: Convert to proper scientific notation: 3.6 × 10¹ × 10⁶ = 3.6 × 10⁷ Step 7: Include units: The volume is 3.6 × 10⁷ cubic meters

  2. (6⁴ × 6⁻²) ÷ (6³ × 6⁻⁴) = ? Answer: 216 Solution: Simplify the numerator: 6⁴ × 6⁻² = 6^(4 + (-2)) = 6² Simplify the denominator: 6³ × 6⁻⁴ = 6^(3 + (-4)) = 6⁻¹ Now divide: 6² ÷ 6⁻¹ = 6^(2 - (-1)) = 6^(2 + 1) = 6³ Calculate 6³ = 6 × 6 × 6 = 216 The answer is 216.
    Full step-by-step solution

    Step 1: Simplify the numerator: 6⁴ × 6⁻² = 6^(4 + (-2)) = 6² Step 2: Simplify the denominator: 6³ × 6⁻⁴ = 6^(3 + (-4)) = 6⁻¹ Step 3: Now divide: 6² ÷ 6⁻¹ = 6^(2 - (-1)) = 6^(2 + 1) = 6³ Step 4: Calculate 6³ = 6 × 6 × 6 = 216 The answer is 216.

  3. (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.

  4. Liam is studying the growth of algae in a pond for his science project. He observes that the algae population starts at 8,000 cells and triples every 2 hours. After 6 hours, a special treatment is applied that reduces the population to one-ninth of its size at that moment. What is the final population of algae cells? Express your answer in standard form. Answer: 24000 Solution: Determine the number of growth cycles in 6 hours. Since the population triples every 2 hours, there are 6 ÷ 2 = 3 growth cycles. Calculate the population after 3 growth cycles.
    Full step-by-step solution

    Step 1: Determine the number of growth cycles in 6 hours. Since the population triples every 2 hours, there are 6 ÷ 2 = 3 growth cycles. Step 2: Calculate the population after 3 growth cycles. Starting population: 8,000 After first cycle: 8,000 × 3 = 24,000 After second cycle: 24,000 × 3 = 72,000 After third cycle: 72,000 × 3 = 216,000 Step 3: Apply the treatment that reduces population to one-ninth: 216,000 ÷ 9 = 24,000 Step 4: The final population is 24,000 cells.

  5. Sophia is designing a rectangular garden for her school's science project. The length of the garden is 4^3 feet and the width is 4^1 feet. She plans to cover the entire garden with soil. What is the area of the garden in square feet? Express your answer as a single power of 4. Answer: 4^4 Solution: The area of a rectangle is length times width. Area = 4^3 * 4^1 Using the property x^m * x^n = x^(m+n), we add the exponents: 4^(3+1) = 4^4 The area of the garden is 4^4 square feet.
    Full step-by-step solution

    Step 1: The area of a rectangle is length times width. Step 2: Area = 4^3 * 4^1 Step 3: Using the property x^m * x^n = x^(m+n), we add the exponents: 4^(3+1) = 4^4 Step 4: The area of the garden is 4^4 square feet. The answer is 4^4.

  6. Matiu is helping his school's technology club design a sound amplifier. The amplifier's power output in watts can be modeled by the expression (3^4 × 3^2)^3. Matiu needs to simplify this expression to find the power output in exponential form with a single base. What is the simplified expression? Answer: 3^18 Solution: Start with the expression (3^4 × 3^2)^3 Apply the product of powers rule inside the parentheses: 3^4 × 3^2 = 3^(4+2) = 3^6 Now we have (3^6)^3 Apply the power of a power rule: (3^6)^3 = 3^(6×3) = 3^18 The simplified expression is 3^18.
    Full step-by-step solution

    Step 1: Start with the expression (3^4 × 3^2)^3 Step 2: Apply the product of powers rule inside the parentheses: 3^4 × 3^2 = 3^(4+2) = 3^6 Step 3: Now we have (3^6)^3 Step 4: Apply the power of a power rule: (3^6)^3 = 3^(6×3) = 3^18 The simplified expression is 3^18.

  7. (9³ × 9⁻⁴) ÷ (9⁻² × 9⁰) = ? Answer: 9 Solution: Simplify the numerator: 9³ × 9⁻⁴ = 9^(3 + (-4)) = 9⁻¹ Simplify the denominator: 9⁻² × 9⁰ = 9^(-2 + 0) = 9⁻² Now we have 9⁻¹ ÷ 9⁻² Apply the quotient of powers property: 9^(-1 - (-2)) = 9^(-1 + 2) = 9¹ 9¹ = 9 The answer is 9.
    Full step-by-step solution

    Step 1: Simplify the numerator: 9³ × 9⁻⁴ = 9^(3 + (-4)) = 9⁻¹ Step 2: Simplify the denominator: 9⁻² × 9⁰ = 9^(-2 + 0) = 9⁻² Step 3: Now we have 9⁻¹ ÷ 9⁻² Step 4: Apply the quotient of powers property: 9^(-1 - (-2)) = 9^(-1 + 2) = 9¹ Step 5: 9¹ = 9 The answer is 9.

  8. (6³ × 6⁻¹) ÷ (6² × 6⁻⁴) = ? Answer: 1296 Solution: Step 1: Apply the product of powers property to the numerator: 6³ × 6⁻¹ = 6^(3 + (-1)) = 6² Step 2: Apply the product of powers property to the denominator: 6² × 6⁻⁴ = 6^(2 + (-4)) = 6⁻² Step 3: Now we have 6² ÷ 6⁻² Step 4: Apply the quotient of powers property: 6^(2 - (-2)) = 6^(2 + 2) = 6⁴…
    Full step-by-step solution

    Step 1: Apply the product of powers property to the numerator: 6³ × 6⁻¹ = 6^(3 + (-1)) = 6² Step 2: Apply the product of powers property to the denominator: 6² × 6⁻⁴ = 6^(2 + (-4)) = 6⁻² Step 3: Now we have 6² ÷ 6⁻² Step 4: Apply the quotient of powers property: 6^(2 - (-2)) = 6^(2 + 2) = 6⁴ Step 5: Calculate 6⁴ = 6 × 6 × 6 × 6 = 1296 The answer is 1296.