Expressions with Exponents

lessonbunny.com

Answer Key & Explanations

Expressions with Exponents · Grade 6 · Worksheet 3

1. Isabella is helping her school's art club create a large mosaic for the hallway wall. The mosaic will be a perfect square with a side length of 27 inches. She needs to calculate the total area of the mosaic to know how many tiles to buy. Each small tile covers 1 square inch. Write an expression using an exponent to represent the area of the mosaic, then evaluate it to find how many tiles Isabella needs. Answer: 729 Solution: The mosaic is a square with side length 27 inches. The area of a square is side length times side length, so the expression is 27 × 27.
   Full step-by-step solution

   Step 1: The mosaic is a square with side length 27 inches. The area of a square is side length times side length, so the expression is 27 × 27. Step 2: Using an exponent, this is written as 27² (27 squared), because the side length is used as a factor twice. Step 3: Evaluate 27²: 27 × 27 = 729. Step 4: So, Isabella needs 729 tiles. The answer is 729.

2. Emma is designing a square garden in her backyard. She draws the garden on a coordinate grid with corners at (0, 0), (15, 0), (15, 15), and (0, 15). Inside the garden, she places a square flower bed whose side length is 5 units. What is the area of the garden that is NOT covered by the flower bed? (Hint: Write an expression using exponents to find the areas.) Answer: 200 Solution: Find the area of the garden. The garden is a square with side length 15 units. Area = 15 squared = 15^2 = 15 x 15 = 225 square units.
   Full step-by-step solution

   Step 1: Find the area of the garden. The garden is a square with side length 15 units. Area = 15 squared = 15^2 = 15 x 15 = 225 square units. Step 2: Find the area of the flower bed. The flower bed is a square with side length 5 units. Area = 5 squared = 5^2 = 5 x 5 = 25 square units. Step 3: Find the area not covered by the flower bed. Subtract the flower bed area from the garden area: 225 - 25 = 200 square units. The answer is 200.

3. Noah is building a square patio using square tiles. The patio has a side length of 9 feet. Each tile is a square with a side length of 1 foot. Noah decides to create a decorative border around the patio by placing a row of special tiles along the outer edge. The rest of the patio will be filled with plain tiles. Write an expression using exponents to represent the number of plain tiles needed, and then evaluate it. Answer: 49 Solution: The patio is a square with side length 9 feet, so the total number of tiles needed is 9 squared, which is 9^2 = 81 tiles. The border is one tile wide all around.
   Full step-by-step solution

   Step 1: The patio is a square with side length 9 feet, so the total number of tiles needed is 9 squared, which is 9^2 = 81 tiles. Step 2: The border is one tile wide all around. That means the inner square (the part without the border) has a side length of 9 - 2 = 7 feet, because you remove 1 foot from each side. Step 3: The number of plain tiles (the inner square) is 7 squared, or 7^2 = 49 tiles. Step 4: So the expression is 7^2, and the value is 49. Final answer: 49.