Approximate Irrationals

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Answer Key & Explanations

Approximate Irrationals · Grade 8 · Worksheet 1

1. √(2) × √(8) = ? Answer: 4 Solution: Write down the original problem. √(2) × √(8) Use the multiplication rule for square roots. The rule says: √(a) × √(b) = √(a × b) So: √(2) × √(8) = √(2 × 8) Multiply the numbers inside the square root.
   Full step-by-step solution

   Step 1: Write down the original problem. We have: √(2) × √(8) Step 2: Use the multiplication rule for square roots. The rule says: √(a) × √(b) = √(a × b) So: √(2) × √(8) = √(2 × 8) Step 3: Multiply the numbers inside the square root. 2 × 8 = 16 So we have: √(16) Step 4: Simplify √(16). We know 16 = 4 × 4, so √(16) = 4. Step 5: State the final answer. Therefore, √(2) × √(8) = 4.

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). What is the length of the hypotenuse? Round your answer to the nearest tenth. Answer: 10.0 Solution: Identify the triangle's vertices and sides. The vertices are (0,0), (6,0), and (6,8). The points (0,0) and (6,0) lie on the x-axis, so the distance between them is the length of one leg.
   Full step-by-step solution

   Step 1: Identify the triangle's vertices and sides. The vertices are (0,0), (6,0), and (6,8). The points (0,0) and (6,0) lie on the x-axis, so the distance between them is the length of one leg. The points (6,0) and (6,8) lie on a vertical line, so the distance between them is the length of the other leg. Step 2: Calculate the lengths of the legs. From (0,0) to (6,0): Change in x = 6 - 0 = 6 Change in y = 0 - 0 = 0 Length = 6 So one leg is 6 units long. From (6,0) to (6,8): Change in x = 6 - 6 = 0 Change in y = 8 - 0 = 8 Length = 8 So the other leg is 8 units long. Step 3: Apply the Pythagorean theorem. The Pythagorean theorem states: a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse. Here, a = 6, b = 8. So: 6^2 + 8^2 = c^2 36 + 64 = c^2 100 = c^2 Step 4: Solve for the hypotenuse. c^2 = 100 Take the square root of both sides: c = sqrt(100) c = 10 Step 5: Round to the nearest tenth. 10.0 is already rounded to the nearest tenth. Final Answer: 10.0

3. √(2) + √(8) = ? Answer: 3√2 Solution: Write the problem clearly. √(2) + √(8) Notice that 8 can be factored into 4 × 2. So √(8) = √(4 × 2) Use the property √(a × b) = √(a) × √(b).
   Full step-by-step solution

   Let's solve step by step. Step 1: Write the problem clearly. √(2) + √(8) Step 2: Notice that 8 can be factored into 4 × 2. So √(8) = √(4 × 2) Step 3: Use the property √(a × b) = √(a) × √(b). √(4 × 2) = √(4) × √(2) Step 4: Simplify √(4). √(4) = 2 So √(8) = 2 × √(2) Step 5: Substitute back into the original expression. √(2) + √(8) = √(2) + 2 × √(2) Step 6: Factor out √(2). √(2) × (1 + 2) = √(2) × 3 Step 7: Write the final answer. 3√(2) So the answer is 3√2.

4. √(75) ≈ ? (to the nearest tenth) Answer: 8.7 Solution: Identify perfect squares near 75. 8^2 = 64 and 9^2 = 81, so √75 is between 8 and 9. 75 is closer to 81 than to 64.
   Full step-by-step solution

   Step 1: Identify perfect squares near 75. 8^2 = 64 and 9^2 = 81, so √75 is between 8 and 9. Step 2: 75 is closer to 81 than to 64. 75 - 64 = 11, 81 - 75 = 6, so it's closer to 9. Step 3: Try 8.6: 8.6^2 = 73.96 (too low) Step 4: Try 8.7: 8.7^2 = 75.69 (slightly high) Step 5: Try 8.65: 8.65^2 = 74.8225 (too low) Step 6: Since 8.7^2 = 75.69 is closer to 75 than 8.6^2 = 73.96, round to 8.7. The answer is 8.7.

5. Hana is creating a square-shaped garden in her backyard. She wants the area of the garden to be exactly 85 square meters so she can plant a specific number of vegetables. To buy fencing for the perimeter, she needs to know the side length of the garden. Since 85 is not a perfect square, she needs to approximate the side length to the nearest tenth of a meter. What is the approximate side length of Hana's garden, rounded to the nearest tenth of a meter? Answer: 9.2 Solution: Find the two perfect squares closest to 85. 9^2 = 81 and 10^2 = 100. So, sqrt(85) is between 9 and 10.
   Full step-by-step solution

   Step 1: Find the two perfect squares closest to 85. 9^2 = 81 and 10^2 = 100. So, sqrt(85) is between 9 and 10. Step 2: Try 9.2. 9.2^2 = 84.64, which is less than 85. Step 3: Try 9.3. 9.3^2 = 86.49, which is greater than 85. Step 4: Since 84.64 is closer to 85 than 86.49 is (85 - 84.64 = 0.36, 86.49 - 85 = 1.49), sqrt(85) is closer to 9.2. Step 5: Therefore, the approximate side length is 9.2 meters.

6. Liam is designing a circular garden with a diameter of 12 meters. He needs to calculate the exact circumference to order the right amount of border fencing. However, the fencing company only accepts measurements rounded to the nearest tenth of a meter. What is the circumference of Liam's garden, rounded to the nearest tenth? (Use π = 3.14) Answer: 37.7 Solution: Recall the formula for circumference. The circumference C of a circle is given by C = π × d, where d is the diameter. Identify the given values.
   Full step-by-step solution

   Step 1: Recall the formula for circumference. The circumference C of a circle is given by C = π × d, where d is the diameter. Step 2: Identify the given values. Diameter d = 12 meters π = 3.14 (as given in the problem) Step 3: Substitute the values into the formula. C = 3.14 × 12 Step 4: Perform the multiplication. 3.14 × 12 = 37.68 Step 5: Round the result to the nearest tenth. 37.68 — look at the hundredths digit, which is 8. Since 8 is greater than or equal to 5, round the tenths digit (6) up by 1. So 37.68 rounded to the nearest tenth is 37.7. Step 6: State the final answer. The circumference rounded to the nearest tenth is 37.7 meters.