Integer Exponents

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Answer Key & Explanations

Integer Exponents · Grade 8 · Worksheet 3

1. Isabella is helping her school's astronomy club build a model of a galaxy. The model shows that a certain star cluster has an initial brightness of 8^4 lumens. As the cluster ages, its brightness increases by a factor of 8^5 every million years. After 3 million years, the cluster undergoes a supernova event that multiplies its current brightness by (8^2)^3. What is the final brightness of the star cluster in exponential form with a base of 8? Answer: 8^25 Solution: Start with the initial brightness: 8^4. After 3 million years, the brightness increases by a factor of 8^5 each million years. Multiply the initial brightness by the growth factor: 8^4 * 8^15 = 8^(4+15) = 8^19.
   Full step-by-step solution

   Step 1: Start with the initial brightness: 8^4. Step 2: After 3 million years, the brightness increases by a factor of 8^5 each million years. So the growth factor is (8^5)^3. Apply the power of a power rule: (8^5)^3 = 8^(5*3) = 8^15. Step 3: Multiply the initial brightness by the growth factor: 8^4 * 8^15 = 8^(4+15) = 8^19. Step 4: The supernova multiplies this result by (8^2)^3. Apply the power of a power rule: (8^2)^3 = 8^(2*3) = 8^6. Step 5: Multiply: 8^19 * 8^6 = 8^(19+6) = 8^25. The final brightness is 8^25 lumens.

2. Emma is designing a rectangular garden for her school project. The length of the garden is 7^3 centimeters and the width is 7^5 centimeters. She wants to cover the entire garden with soil. What is the area of the garden in square centimeters? Express your answer as a power of 7. Answer: 7^8 Solution: The area of a rectangle is length times width. So, area = 7^3 * 7^5. Use the property of exponents: when multiplying powers with the same base, add the exponents: 7^3 * 7^5 = 7^(3+5).
   Full step-by-step solution

   Step 1: The area of a rectangle is length times width. So, area = 7^3 * 7^5. Step 2: Use the property of exponents: when multiplying powers with the same base, add the exponents: 7^3 * 7^5 = 7^(3+5). Step 3: Add the exponents: 3 + 5 = 8. Step 4: Therefore, the area is 7^8 square centimeters. The answer is 7^8.

3. (8³ × 4⁻²) ÷ (2⁵ × 2⁻³) = ? Answer: 8 Solution: Write all terms with base 2. 8 = 2³, so 8³ = (2³)³ = 2⁹. 4 = 2², so 4⁻² = (2²)⁻² = 2⁻⁴.
   Full step-by-step solution

   Step 1: Write all terms with base 2. 8 = 2³, so 8³ = (2³)³ = 2⁹. 4 = 2², so 4⁻² = (2²)⁻² = 2⁻⁴. Step 2: Rewrite the expression: (2⁹ × 2⁻⁴) ÷ (2⁵ × 2⁻³). Step 3: Simplify numerator: 2⁹ × 2⁻⁴ = 2^(9 + (-4)) = 2⁵. Step 4: Simplify denominator: 2⁵ × 2⁻³ = 2^(5 + (-3)) = 2². Step 5: Divide: 2⁵ ÷ 2² = 2^(5 - 2) = 2³. Step 6: Calculate 2³ = 8. The answer is 8.

4. (5³ × 5⁻²) ÷ 5² = ? Answer: 0.2 Solution: Step 1: Apply the product of powers property to the numerator: 5³ × 5⁻² = 5^(3 + (-2)) = 5¹ Step 2: Now we have 5¹ ÷ 5² Step 3: Apply the quotient of powers property: 5¹ ÷ 5² = 5^(1 - 2) = 5⁻¹ Step 4: Rewrite the negative exponent as a reciprocal: 5⁻¹ = 1/5¹ = 1/5 Step 5: Convert to decimal: 1/5…
   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: Now we have 5¹ ÷ 5² Step 3: Apply the quotient of powers property: 5¹ ÷ 5² = 5^(1 - 2) = 5⁻¹ Step 4: Rewrite the negative exponent as a reciprocal: 5⁻¹ = 1/5¹ = 1/5 Step 5: Convert to decimal: 1/5 = 0.2 The answer is 0.2.

5. (5³ × 5⁻²) ÷ (5⁴ × 5⁻⁵) = ? Answer: 25 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: 5⁴ × 5⁻⁵ = 5^(4 + (-5)) = 5⁻¹ Step 3: Now we have 5¹ ÷ 5⁻¹ Step 4: Apply the quotient of powers property: 5^(1 - (-1)) = 5^(1 + 1) = 5²…
   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: 5⁴ × 5⁻⁵ = 5^(4 + (-5)) = 5⁻¹ Step 3: Now we have 5¹ ÷ 5⁻¹ Step 4: Apply the quotient of powers property: 5^(1 - (-1)) = 5^(1 + 1) = 5² Step 5: Calculate 5² = 25 The answer is 25.

6. Emma is helping her school's science club measure the volume of a large rectangular storage tank. The tank has a length of 5^4 centimeters, a width of 5^2 centimeters, and a height of 5^3 centimeters. What is the volume of the tank in cubic centimeters? Express your answer as a single power of 5. Answer: 5^9 Solution: The volume of a rectangular prism is length times width times height. Volume = 5^4 × 5^2 × 5^3 Using the property of integer exponents (x^m × x^n = x^(m+n)), we add the exponents since the base is the same (5).
   Full step-by-step solution

   Step 1: The volume of a rectangular prism is length times width times height. Step 2: Volume = 5^4 × 5^2 × 5^3 Step 3: Using the property of integer exponents (x^m × x^n = x^(m+n)), we add the exponents since the base is the same (5). Step 4: Volume = 5^(4+2+3) = 5^9 Step 5: The volume of the tank is 5^9 cubic centimeters. The answer is 5^9.

7. Tane is helping his school's science club build a model of a solar system. The volume of a spherical model planet is given by the expression (5^3)^4 cubic centimeters. Tane needs to simplify this expression to find the volume as a single power of 5. What is the simplified expression? Answer: 5^12 Solution: Start with the expression (5^3)^4 Apply the power of a power rule: (x^m)^n = x^(m*n) Multiply the exponents: 3 * 4 = 12 The simplified expression is 5^12 The answer is 5^12.
   Full step-by-step solution

   Step 1: Start with the expression (5^3)^4 Step 2: Apply the power of a power rule: (x^m)^n = x^(m*n) Step 3: Multiply the exponents: 3 * 4 = 12 Step 4: The simplified expression is 5^12 The answer is 5^12.

8. 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.