Integer Exponents

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Answer Key & Explanations

Integer Exponents · Grade 8 · Worksheet 2

1. (3⁵ × 3⁻³) ÷ (3² × 3⁻¹) = ? Answer: 3 Solution: Simplify the numerator: 3⁵ × 3⁻³ = 3^(5 + (-3)) = 3² Simplify the denominator: 3² × 3⁻¹ = 3^(2 + (-1)) = 3¹ Divide: 3² ÷ 3¹ = 3^(2 - 1) = 3¹ 3¹ = 3 The answer is 3.
   Full step-by-step solution

   Step 1: Simplify the numerator: 3⁵ × 3⁻³ = 3^(5 + (-3)) = 3² Step 2: Simplify the denominator: 3² × 3⁻¹ = 3^(2 + (-1)) = 3¹ Step 3: Divide: 3² ÷ 3¹ = 3^(2 - 1) = 3¹ Step 4: 3¹ = 3 The answer is 3.

2. (7⁵ × 7⁻³) ÷ (7² × 7⁻⁴) = ? Answer: 2401 Solution: Simplify the numerator: 7⁵ × 7⁻³ = 7^(5 + (-3)) = 7² Simplify the denominator: 7² × 7⁻⁴ = 7^(2 + (-4)) = 7⁻² Now we have 7² ÷ 7⁻² = 7^(2 - (-2)) = 7^(2 + 2) = 7⁴ Calculate 7⁴ = 7 × 7 × 7 × 7 = 49 × 49 = 2401 The answer is 2401.
   Full step-by-step solution

   Step 1: Simplify the numerator: 7⁵ × 7⁻³ = 7^(5 + (-3)) = 7² Step 2: Simplify the denominator: 7² × 7⁻⁴ = 7^(2 + (-4)) = 7⁻² Step 3: Now we have 7² ÷ 7⁻² = 7^(2 - (-2)) = 7^(2 + 2) = 7⁴ Step 4: Calculate 7⁴ = 7 × 7 × 7 × 7 = 49 × 49 = 2401 The answer is 2401.

3. Aroha is helping her school's science club build a model of a solar system. They have a scale where the distance from the Sun to Neptune is represented as 7.2 × 10⁹ kilometers. The distance from the Sun to Earth is 1.5 × 10⁸ kilometers. How many times farther is Neptune from the Sun compared to Earth? Express your answer as a whole number. Answer: 48 Solution: Write the division: (7.2 × 10⁹) ÷ (1.5 × 10⁸). Separate the coefficients and powers of 10: (7.2 ÷ 1.5) × (10⁹ ÷ 10⁸). Divide the coefficients: 7.2 ÷ 1.5 = 4.8.
   Full step-by-step solution

   Step 1: Write the division: (7.2 × 10⁹) ÷ (1.5 × 10⁸). Step 2: Separate the coefficients and powers of 10: (7.2 ÷ 1.5) × (10⁹ ÷ 10⁸). Step 3: Divide the coefficients: 7.2 ÷ 1.5 = 4.8. Step 4: Apply the quotient rule for exponents: 10⁹ ÷ 10⁸ = 10^(9-8) = 10¹ = 10. Step 5: Multiply: 4.8 × 10 = 48. Step 6: The answer is 48. Neptune is 48 times farther from the Sun than Earth.

4. A rectangular prism has dimensions of 3 × 10² cm by 2 × 10³ cm by 5 × 10¹ cm. Using the properties of exponents, calculate the volume of the prism in scientific notation. Answer: 3 × 10⁷ cm³ Solution: Write down the volume formula for a rectangular prism. Volume = length × width × height. Substitute the given dimensions into the formula.
   Full step-by-step solution

   Step 1: Write down the volume formula for a rectangular prism. Volume = length × width × height. Step 2: Substitute the given dimensions into the formula. Length = 3 × 10² cm Width = 2 × 10³ cm Height = 5 × 10¹ cm So: Volume = (3 × 10²) × (2 × 10³) × (5 × 10¹) Step 3: Group the coefficients (regular numbers) together and the powers of 10 together. Coefficients: 3 × 2 × 5 Powers of 10: 10² × 10³ × 10¹ Step 4: Multiply the coefficients. 3 × 2 = 6 6 × 5 = 30 So coefficients multiply to 30. Step 5: Multiply the powers of 10 using the exponent rule: a^m × a^n = a^(m+n). 10² × 10³ × 10¹ = 10^(2 + 3 + 1) = 10⁶ Step 6: Combine the results. Volume = 30 × 10⁶ cm³ Step 7: Convert 30 × 10⁶ into proper scientific notation (a number between 1 and 10 times a power of 10). 30 = 3.0 × 10¹ So 30 × 10⁶ = (3.0 × 10¹) × 10⁶ = 3.0 × 10^(1 + 6) = 3.0 × 10⁷ Step 8: Write the final answer with units. Volume = 3 × 10⁷ cm³

5. Mere is examining a cube-shaped sculpture. Each side length of the cube is 4 × 10^3 millimeters. Using the properties of integer exponents, calculate the volume of the cube in cubic millimeters and express your answer in scientific notation. Answer: 6.4 × 10^10 Solution: Write the volume formula for a cube: Volume = side^3. Substitute the given side length: Volume = (4 × 10^3)^3. So (4 × 10^3)^3 = 4^3 × (10^3)^3.
   Full step-by-step solution

   Step 1: Write the volume formula for a cube: Volume = side^3. Step 2: Substitute the given side length: Volume = (4 × 10^3)^3. Step 3: Apply the power of a product property: (a × b)^n = a^n × b^n. So (4 × 10^3)^3 = 4^3 × (10^3)^3. Step 4: Calculate 4^3 = 4 × 4 × 4 = 64. Step 5: Apply the power of a power property: (10^3)^3 = 10^(3×3) = 10^9. Step 6: Combine the results: 64 × 10^9. Step 7: Convert to proper scientific notation: 64 × 10^9 = 6.4 × 10^1 × 10^9 = 6.4 × 10^(1+9) = 6.4 × 10^10. The answer is 6.4 × 10^10.

6. Aroha is designing a square garden with side length 9 × 10² meters. She wants to place a smaller square pond in the center with side length 3 × 10¹ meters. Using properties of exponents, find the area of the garden that is NOT covered by the pond (the remaining area) in scientific notation. Answer: 8.1 × 10⁵ Solution: Find the area of the garden. Area of garden = (side length)² = (9 × 10²)² = 9² × (10²)² = 81 × 10⁴ = 8.1 × 10⁵ square meters. Find the area of the pond.
   Full step-by-step solution

   Step 1: Find the area of the garden. Area of garden = (side length)² = (9 × 10²)² = 9² × (10²)² = 81 × 10⁴ = 8.1 × 10⁵ square meters. Step 2: Find the area of the pond. Area of pond = (3 × 10¹)² = 3² × (10¹)² = 9 × 10² = 0.09 × 10⁵ square meters. Step 3: Subtract the pond area from the garden area. 8.1 × 10⁵ - 0.09 × 10⁵ = (8.1 - 0.09) × 10⁵ = 8.01 × 10⁵ square meters. The answer is 8.01 × 10⁵.

7. (8⁴ × 8⁻²) ÷ (8³ × 8⁻⁶) = ? Answer: 32768 Solution: Simplify the numerator: 8⁴ × 8⁻² = 8^(4 + (-2)) = 8² Simplify the denominator: 8³ × 8⁻⁶ = 8^(3 + (-6)) = 8⁻³ Now we have 8² ÷ 8⁻³ = 8^(2 - (-3)) = 8^(2 + 3) = 8⁵ Calculate 8⁵ = 8 × 8 × 8 × 8 × 8 = 64 × 64 × 8 = 4096 × 8 = 32768 The answer is 32768.
   Full step-by-step solution

   Step 1: Simplify the numerator: 8⁴ × 8⁻² = 8^(4 + (-2)) = 8² Step 2: Simplify the denominator: 8³ × 8⁻⁶ = 8^(3 + (-6)) = 8⁻³ Step 3: Now we have 8² ÷ 8⁻³ = 8^(2 - (-3)) = 8^(2 + 3) = 8⁵ Step 4: Calculate 8⁵ = 8 × 8 × 8 × 8 × 8 = 64 × 64 × 8 = 4096 × 8 = 32768 The answer is 32768.

8. Liam is studying the growth of algae in a pond. The algae population starts with 5 × 10³ cells and quadruples every day. After 3 days, a chemical treatment is applied that reduces the population to one-sixteenth of its current size. What is the final algae population in scientific notation? Answer: 3.2 × 10^4 Solution: Initial population is 5 × 10³ cells After 3 days of quadrupling, population = 5 × 10³ × 4³ Calculate 4³ = 64 Population after 3 days = 5 × 10³ × 64 = 320 × 10³ = 3.2 × 10^5 Chemical treatment reduces population to 1/16 of current size Final population = 3.2 × 10^5 ÷ 16 = 3.2 × 10^5 ÷ 2^4 3.2 ÷…
   Full step-by-step solution

   Step 1: Initial population is 5 × 10³ cells Step 2: After 3 days of quadrupling, population = 5 × 10³ × 4³ Step 3: Calculate 4³ = 64 Step 4: Population after 3 days = 5 × 10³ × 64 = 320 × 10³ = 3.2 × 10^5 Step 5: Chemical treatment reduces population to 1/16 of current size Step 6: Final population = 3.2 × 10^5 ÷ 16 = 3.2 × 10^5 ÷ 2^4 Step 7: 3.2 ÷ 16 = 0.2, so 0.2 × 10^5 = 2 × 10^4 Step 8: The final population is 3.2 × 10^4 cells