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Expressions with Exponents

Grade 6 · Algebra · Worksheet 1

  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 hall and the fire station is 450 meters, how many centimeters apart should these two buildings be placed on Liam's model? Answer: ______________
  2. Liam is creating a scale model of his neighborhood where 1 centimeter represents 15 meters in real life. If the actual distance between the library and the school is 225 meters, how many centimeters apart should these two buildings be placed on Liam's model? Answer: ______________
  3. Maya is planning a community garden and needs to calculate how much soil to order. The garden area is rectangular, measuring 15.4 meters long by 8.2 meters wide. She needs soil to a depth of 0.3 meters. What volume of soil, in cubic meters, does Maya need to order for the garden? Answer: ______________
  4. Liam is designing a square mosaic tile for an art competition. He draws a large square on a coordinate grid with corners at (1, 1), (10, 1), (10, 10), and (1, 10). Inside this large square, he places a smaller square at the center, with a side length of 3 units. Liam wants to write an expression using exponents to represent the area of the large square that is NOT covered by the small square. What is the value of this expression?
    Answer: ______________
  5. Noah is organizing the school's science fair. He needs to arrange 6 display tables in each row. If he makes 6 rows of tables, how many tables will there be in total? Write the expression using an exponent and then evaluate it. Answer: ______________
  6. Mason is designing a square tile mosaic. The side length of the square mosaic is 9 tiles. He then decides to add a smaller square in the center with a side length of 4 tiles. Write an expression using exponents to represent the total number of tiles in the large square minus the tiles in the small square, then evaluate it to find how many tiles are in the border region. Answer: ______________
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Answer Key & Explanations

Expressions with Exponents · Grade 6 · Worksheet 1

  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 hall and the fire station is 450 meters, how many centimeters apart should these two buildings be placed on Liam's model? Answer: 12 Solution: Identify the scale ratio: 2 cm represents 75 meters Set up a proportion: model distance / actual distance = 2 cm / 75 meters Let x be the model distance in centimeters: x / 450 = 2 / 75 Cross multiply: 75x = 2 × 450 Calculate: 75x = 900 Solve for x: x = 900 ÷ 75 Calculate: x = 12 The answer is…
    Full step-by-step solution

    Step 1: Identify the scale ratio: 2 cm represents 75 meters Step 2: Set up a proportion: model distance / actual distance = 2 cm / 75 meters Step 3: Let x be the model distance in centimeters: 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. Liam is creating a scale model of his neighborhood where 1 centimeter represents 15 meters in real life. If the actual distance between the library and the school is 225 meters, how many centimeters apart should these two buildings be placed on Liam's model? Answer: 15 Solution: The scale is 1 cm on the model = 15 meters in real life. Identify the actual distance. The actual distance between the library and the school is 225 meters.
    Full step-by-step solution

    Step 1: Understand the scale. The scale is 1 cm on the model = 15 meters in real life. Step 2: Identify the actual distance. The actual distance between the library and the school is 225 meters. Step 3: Set up the relationship. Let the model distance in centimeters be \( x \). From the scale: 1 cm / 15 m = \( x \) cm / 225 m Step 4: Solve for \( x \). Multiply both sides by 225: \( x = (225 / 15) \) cm Step 5: Perform the division. 225 ÷ 15 = 15 Step 6: State the answer. The model distance is 15 cm.

  3. Maya is planning a community garden and needs to calculate how much soil to order. The garden area is rectangular, measuring 15.4 meters long by 8.2 meters wide. She needs soil to a depth of 0.3 meters. What volume of soil, in cubic meters, does Maya need to order for the garden? Answer: 37.884 Solution: Identify the dimensions: length = 15.4 m, width = 8.2 m, depth = 0.3 m Calculate the volume using the formula: volume = length × width × depth Multiply length and width: 15.4 × 8.2 = 126.28 Multiply the result by the depth: 126.28 × 0.3 = 37.884 The volume of soil needed is 37.884 cubic meters.
    Full step-by-step solution

    Step 1: Identify the dimensions: length = 15.4 m, width = 8.2 m, depth = 0.3 m Step 2: Calculate the volume using the formula: volume = length × width × depth Step 3: Multiply length and width: 15.4 × 8.2 = 126.28 Step 4: Multiply the result by the depth: 126.28 × 0.3 = 37.884 Step 5: The volume of soil needed is 37.884 cubic meters.

  4. Liam is designing a square mosaic tile for an art competition. He draws a large square on a coordinate grid with corners at (1, 1), (10, 1), (10, 10), and (1, 10). Inside this large square, he places a smaller square at the center, with a side length of 3 units. Liam wants to write an expression using exponents to represent the area of the large square that is NOT covered by the small square. What is the value of this expression? Answer: 72 Solution: Find the side length of the large square. The corners are at (1, 1) and (10, 1). The horizontal distance is 10 - 1 = 9 units.
    Full step-by-step solution

    Step 1: Find the side length of the large square. The corners are at (1, 1) and (10, 1). The horizontal distance is 10 - 1 = 9 units. So side length = 9 units. Step 2: Find the area of the large square. Area = 9 squared = 9^2 = 9 x 9 = 81 square units. Step 3: Find the area of the small square. Side length = 3 units. Area = 3 squared = 3^2 = 3 x 3 = 9 square units. Step 4: Find the area not covered. Uncovered area = 81 - 9 = 72 square units. The answer is 72.

  5. Noah is organizing the school's science fair. He needs to arrange 6 display tables in each row. If he makes 6 rows of tables, how many tables will there be in total? Write the expression using an exponent and then evaluate it. Answer: 36 Solution: There are 6 tables in each row and 6 rows. This means we multiply 6 by 6. The multiplication 6 × 6 can be written as 6 squared, which is 6^2.
    Full step-by-step solution

    Step 1: There are 6 tables in each row and 6 rows. This means we multiply 6 by 6. Step 2: The multiplication 6 × 6 can be written as 6 squared, which is 6^2. Step 3: Evaluate: 6^2 = 6 × 6 = 36. The total number of tables is 36.

  6. Mason is designing a square tile mosaic. The side length of the square mosaic is 9 tiles. He then decides to add a smaller square in the center with a side length of 4 tiles. Write an expression using exponents to represent the total number of tiles in the large square minus the tiles in the small square, then evaluate it to find how many tiles are in the border region. Answer: 65 Solution: Write the expression for the large square area: 9^2. Write the expression for the small square area: 4^2. The border area is the difference: 9^2 - 4^2.
    Full step-by-step solution

    Step 1: Write the expression for the large square area: 9^2. Step 2: Write the expression for the small square area: 4^2. Step 3: The border area is the difference: 9^2 - 4^2. Step 4: Evaluate: 9^2 = 81, 4^2 = 16. Step 5: Subtract: 81 - 16 = 65. The border region contains 65 tiles.