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Expressions with Exponents

Grade 6 · Algebra · Worksheet 3

  1. Liam is creating a scale model of his city where 2 centimeters represents 75 meters in real life. If the actual distance between the city park and the museum is 450 meters, how many centimeters apart should these two landmarks be placed on Liam's model? Answer: ______________
  2. (-5)² - 3 × (12 - 15) ÷ 3 = ? Answer: ______________
  3. Mason is building a square-shaped community garden in his neighborhood park. Each side of the garden will be 12 meters long. To find the total area, Mason calculates the side length squared. However, the park committee also wants a smaller square flower bed inside the garden with a side length of 7 meters. Mason needs to know how much larger the area of the garden is than the area of the flower bed. What is the difference in their areas?
    Answer: ______________
  4. (-3)² + 4 × (15 - 20) ÷ 2 = ? Answer: ______________
  5. A square mosaic tile has a side length of 11 cm. A smaller square pattern inside the tile has a side length of 4 cm. Tane wants to evaluate the area of the larger square minus the area of the smaller square. Write an expression using exponents for this difference and evaluate it.
    Answer: ______________
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Answer Key & Explanations

Expressions with Exponents · Grade 6 · Worksheet 3

  1. Liam is creating a scale model of his city where 2 centimeters represents 75 meters in real life. If the actual distance between the city park and the museum is 450 meters, how many centimeters apart should these two landmarks be placed on Liam's model? Answer: 12 Solution: Identify the scale ratio: 2 cm on model = 75 m in real life Set up a proportion: model distance / real distance = scale ratio Let x be the model distance in cm: x / 450 = 2 / 75 Cross multiply: 75x = 2 × 450 Calculate: 75x = 900 Solve for x: x = 900 ÷ 75 Calculate: x = 12 The answer is 12…
    Full step-by-step solution

    Step 1: Identify the scale ratio: 2 cm on model = 75 m in real life Step 2: Set up a proportion: model distance / real distance = scale ratio Step 3: Let x be the model distance in cm: x / 450 = 2 / 75 Step 4: Cross multiply: 75x = 2 × 450 Step 5: Calculate: 75x = 900 Step 6: Solve for x: x = 900 ÷ 75 Step 7: Calculate: x = 12 The answer is 12 centimeters.

  2. (-5)² - 3 × (12 - 15) ÷ 3 = ? Answer: 28 Solution: Calculate the exponent: (-5)² = 25 Calculate inside parentheses: (12 - 15) = -3 Perform multiplication: 3 × (-3) = -9 Perform division: -9 ÷ 3 = -3 Subtract: 25 - (-3) = 25 + 3 = 28 The answer is 28.
    Full step-by-step solution

    Step 1: Calculate the exponent: (-5)² = 25 Step 2: Calculate inside parentheses: (12 - 15) = -3 Step 3: Perform multiplication: 3 × (-3) = -9 Step 4: Perform division: -9 ÷ 3 = -3 Step 5: Subtract: 25 - (-3) = 25 + 3 = 28 The answer is 28.

  3. Mason is building a square-shaped community garden in his neighborhood park. Each side of the garden will be 12 meters long. To find the total area, Mason calculates the side length squared. However, the park committee also wants a smaller square flower bed inside the garden with a side length of 7 meters. Mason needs to know how much larger the area of the garden is than the area of the flower bed. What is the difference in their areas? Answer: 95 Solution: Find the area of the garden. Since it is a square with side length 12 meters, area = 12 squared = 12^2 = 12 × 12 = 144 square meters. Find the area of the flower bed.
    Full step-by-step solution

    Step 1: Find the area of the garden. Since it is a square with side length 12 meters, area = 12 squared = 12^2 = 12 × 12 = 144 square meters. Step 2: Find the area of the flower bed. It is a square with side length 7 meters, area = 7 squared = 7^2 = 7 × 7 = 49 square meters. Step 3: Find the difference in area: 144 - 49 = 95 square meters. The garden's area is 95 square meters larger than the flower bed's area.

  4. (-3)² + 4 × (15 - 20) ÷ 2 = ? Answer: -1 Solution: Start with the expression: (-3)² + 4 × (15 - 20) ÷ 2 Calculate inside the parentheses: 15 - 20 = -5 The expression becomes: (-3)² + 4 × (-5) ÷ 2 Calculate the exponent: (-3)² = 9 The expression becomes: 9 + 4 × (-5) ÷ 2 Perform multiplication and division from left to right: 4 × (-5) = -20 The…
    Full step-by-step solution

    Step 1: Start with the expression: (-3)² + 4 × (15 - 20) ÷ 2 Step 2: Calculate inside the parentheses: 15 - 20 = -5 Step 3: The expression becomes: (-3)² + 4 × (-5) ÷ 2 Step 4: Calculate the exponent: (-3)² = 9 Step 5: The expression becomes: 9 + 4 × (-5) ÷ 2 Step 6: Perform multiplication and division from left to right: 4 × (-5) = -20 Step 7: The expression becomes: 9 + (-20) ÷ 2 Step 8: Continue with division: (-20) ÷ 2 = -10 Step 9: The expression becomes: 9 + (-10) Step 10: Perform the addition: 9 + (-10) = -1 The answer is -1.

  5. A square mosaic tile has a side length of 11 cm. A smaller square pattern inside the tile has a side length of 4 cm. Tane wants to evaluate the area of the larger square minus the area of the smaller square. Write an expression using exponents for this difference and evaluate it. Answer: 105 Solution: The area of the larger square is side length times side length = 11 cm * 11 cm = 11 squared = 11^2. The area of the smaller square is side length times side length = 4 cm * 4 cm = 4 squared = 4^2.
    Full step-by-step solution

    Step 1: The area of the larger square is side length times side length = 11 cm * 11 cm = 11 squared = 11^2. Step 2: The area of the smaller square is side length times side length = 4 cm * 4 cm = 4 squared = 4^2. Step 3: The difference is 11^2 - 4^2. Step 4: Evaluate: 11^2 = 11 * 11 = 121. Step 5: Evaluate: 4^2 = 4 * 4 = 16. Step 6: Subtract: 121 - 16 = 105. The answer is 105.