Integer Exponents

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Answer Key & Explanations

Integer Exponents · Grade 8 · Worksheet 1

1. (5³ × 2⁴) ÷ (5² × 2²) = ? Answer: 20 Solution: Step 1: Apply the quotient of powers property: a^m ÷ a^n = a^(m-n) Step 2: For base 5: 5³ ÷ 5² = 5^(3-2) = 5¹ = 5 Step 3: For base 2: 2⁴ ÷ 2² = 2^(4-2) = 2² = 4 Step 4: Multiply the results: 5 × 4 = 20 The answer is 20.
   Full step-by-step solution

   Step 1: Apply the quotient of powers property: a^m ÷ a^n = a^(m-n) Step 2: For base 5: 5³ ÷ 5² = 5^(3-2) = 5¹ = 5 Step 3: For base 2: 2⁴ ÷ 2² = 2^(4-2) = 2² = 4 Step 4: Multiply the results: 5 × 4 = 20 The answer is 20.

2. Aroha is designing a series of nested cubes for an art project. The side length of the smallest cube is 3^2 cm. The side length of the next cube is (3^2)^3 cm, and the side length of the largest cube is 3^7 cm. Aroha wants to know how many times the volume of the largest cube is compared to the volume of the smallest cube. What is that ratio? Express your answer as a single power of 3. Answer: 3^9 Solution: The side length of the smallest cube is 3^2 cm. Volume of smallest cube = (3^2)^3 = 3^(2*3) = 3^6 cubic cm. The side length of the largest cube is 3^7 cm.
   Full step-by-step solution

   Step 1: The side length of the smallest cube is 3^2 cm. Volume of smallest cube = (3^2)^3 = 3^(2*3) = 3^6 cubic cm. Step 2: The side length of the largest cube is 3^7 cm. Volume of largest cube = (3^7)^3 = 3^(7*3) = 3^21 cubic cm. Step 3: Ratio of largest volume to smallest volume = 3^21 / 3^6 = 3^(21-6) = 3^15. Step 4: But wait, the problem asks for the ratio of the largest cube's volume to the smallest cube's volume. However, the middle cube's side length is given as (3^2)^3 = 3^6 cm. The largest cube is 3^7 cm. The ratio of volumes is 3^21 / 3^6 = 3^15. But re-reading the problem: the side length of the largest cube is 3^7, and the smallest is 3^2. The ratio of volumes is (3^7)^3 / (3^2)^3 = 3^21 / 3^6 = 3^15. The answer is 3^15. The answer is 3^15.

3. 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.

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

   Step 1: Write down the formula for the volume of a rectangular prism. Volume = length × width × height Step 2: Substitute the given values into the formula. Length = 4 × 10² cm Width = 2 × 10³ cm Height = 5 × 10¹ cm Volume = (4 × 10²) × (2 × 10³) × (5 × 10¹) Step 3: Group the coefficients (regular numbers) and the powers of 10 separately. Coefficients: 4 × 2 × 5 Powers of 10: 10² × 10³ × 10¹ Step 4: Multiply the coefficients. 4 × 2 = 8 8 × 5 = 40 So, coefficients multiply to 40. Step 5: Multiply the powers of 10 using the exponent rule: when multiplying powers with the same base, add the exponents. 10² × 10³ × 10¹ = 10^(2 + 3 + 1) = 10⁶ Step 6: Combine the results from Step 4 and Step 5. Volume = 40 × 10⁶ cm³ Step 7: Convert 40 × 10⁶ into proper scientific notation. In scientific notation, the coefficient must be between 1 and 10. 40 = 4.0 × 10¹ So, 40 × 10⁶ = (4.0 × 10¹) × 10⁶ = 4.0 × 10^(1 + 6) = 4 × 10⁷ cm³ Final Answer: 4 × 10⁷ cm³

5. (4³ × 8²) ÷ (2⁵ × 4²) = ? Answer: 8 Solution: Write all terms with base 2. 4 = 2², so 4³ = (2²)³ = 2⁶ and 4² = (2²)² = 2⁴. 8 = 2³, so 8² = (2³)² = 2⁶.
   Full step-by-step solution

   Step 1: Write all terms with base 2. 4 = 2², so 4³ = (2²)³ = 2⁶ and 4² = (2²)² = 2⁴. 8 = 2³, so 8² = (2³)² = 2⁶. Step 2: Rewrite the expression: (2⁶ × 2⁶) ÷ (2⁵ × 2⁴). Step 3: Simplify numerator: 2⁶ × 2⁶ = 2^(6+6) = 2¹². Step 4: Simplify denominator: 2⁵ × 2⁴ = 2^(5+4) = 2⁹. Step 5: Divide: 2¹² ÷ 2⁹ = 2^(12-9) = 2³. Step 6: Calculate 2³ = 8. The answer is 8.

6. A research satellite is collecting data from a distant galaxy. The satellite transmits data at a rate of 3.2 × 10^8 bytes per second. If it transmits continuously for 2.5 × 10^3 seconds, how many total bytes of data are transmitted? Express your answer in standard scientific notation (a × 10^b where 1 ≤ a < 10). Answer: 8.0 × 10^11 Solution: Multiply the coefficients: 3.2 × 2.5 = 8.0 Add the exponents: 10^(8 + 3) = 10^11 Combine the results: 8.0 × 10^11 Check that the coefficient is between 1 and 10 (8.0 satisfies this requirement) The final answer is 8.0 × 10^11 bytes.
   Full step-by-step solution

   Step 1: Multiply the coefficients: 3.2 × 2.5 = 8.0 Step 2: Add the exponents: 10^(8 + 3) = 10^11 Step 3: Combine the results: 8.0 × 10^11 Step 4: Check that the coefficient is between 1 and 10 (8.0 satisfies this requirement) The final answer is 8.0 × 10^11 bytes.

7. Hana is analyzing the growth of a crystal in her science experiment. The crystal's volume starts at 4 × 10^2 cubic millimeters and triples every hour. After 2 hours, she applies a special coating that multiplies the current volume by 16. What is the final volume of the crystal in cubic millimeters? Express your answer in scientific notation. Answer: 5.76 × 10^4 Solution: Initial volume = 4 × 10^2 cubic mm After 1 hour, volume = 4 × 10^2 × 3 After 2 hours, volume = 4 × 10^2 × 3^2 = 4 × 10^2 × 9 Volume after 2 hours = 36 × 10^2 = 3.6 × 10^3 cubic mm Coating multiplies by 16, so final volume = 3.6 × 10^3 × 16 3.6 × 16 = 57.6, so final volume = 57.6 × 10^3 = 5.76 ×…
   Full step-by-step solution

   Step 1: Initial volume = 4 × 10^2 cubic mm Step 2: After 1 hour, volume = 4 × 10^2 × 3 Step 3: After 2 hours, volume = 4 × 10^2 × 3^2 = 4 × 10^2 × 9 Step 4: Volume after 2 hours = 36 × 10^2 = 3.6 × 10^3 cubic mm Step 5: Coating multiplies by 16, so final volume = 3.6 × 10^3 × 16 Step 6: 3.6 × 16 = 57.6, so final volume = 57.6 × 10^3 = 5.76 × 10^4 cubic mm The final volume is 5.76 × 10^4 cubic mm.

8. (2⁷ × 2⁻³) ÷ (2² × 2⁻⁶) = ? Answer: 256 Solution: Apply product of powers in numerator: 2⁷ × 2⁻³ = 2^(7 + (-3)) = 2⁴ Apply product of powers in denominator: 2² × 2⁻⁶ = 2^(2 + (-6)) = 2⁻⁴ Now we have 2⁴ ÷ 2⁻⁴ Apply quotient of powers: 2^(4 - (-4)) = 2^(4 + 4) = 2⁸ Calculate 2⁸ = 256 The answer is 256.
   Full step-by-step solution

   Step 1: Apply product of powers in numerator: 2⁷ × 2⁻³ = 2^(7 + (-3)) = 2⁴ Step 2: Apply product of powers in denominator: 2² × 2⁻⁶ = 2^(2 + (-6)) = 2⁻⁴ Step 3: Now we have 2⁴ ÷ 2⁻⁴ Step 4: Apply quotient of powers: 2^(4 - (-4)) = 2^(4 + 4) = 2⁸ Step 5: Calculate 2⁸ = 256 The answer is 256.