Rational vs Irrational

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Answer Key & Explanations

Rational vs Irrational · Grade 8 · Worksheet 2

1. √(2 + √(9)) = ? Answer: √5 Solution: √(2 + √(9)) Simplify inside the square roots, starting with the innermost one. √(9) = 3, because 3 × 3 = 9.
   Full step-by-step solution

   Let's solve step by step. Step 1: Start with the expression √(2 + √(9)) Step 2: Simplify inside the square roots, starting with the innermost one. √(9) = 3, because 3 × 3 = 9. Step 3: Replace √(9) with 3 in the original expression: √(2 + 3) Step 4: Add the numbers inside the outer square root: 2 + 3 = 5 Step 5: Now we have: √(5) Step 6: Since 5 is not a perfect square, it remains as √5. Final answer: √5

2. A right triangle is drawn on a coordinate plane with vertices at (0,0), (6,0), and (6,8). The hypotenuse of this triangle is used as the diameter of a circle. What is the exact area of the circle? Express your answer in terms of π. Answer: 25π Solution: A = (0,0) B = (6,0) C = (6,8) This is a right triangle with right angle at B because AB is horizontal and BC is vertical. The hypotenuse is AC, between (0,0) and (6,8).
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Identify the triangle's vertices and hypotenuse** Vertices: A = (0,0) B = (6,0) C = (6,8) This is a right triangle with right angle at B because AB is horizontal and BC is vertical. --- **Step 2: Find the hypotenuse** The hypotenuse is AC, between (0,0) and (6,8). Length AC = sqrt( (6-0)^2 + (8-0)^2 ) = sqrt(36 + 64) = sqrt(100) = 10. So the hypotenuse length is 10. --- **Step 3: Interpret the circle's diameter** The problem says: "The hypotenuse of this triangle is used as the diameter of a circle." So the diameter of the circle = length of AC = 10. --- **Step 4: Find the radius** Radius r = diameter / 2 = 10 / 2 = 5. --- **Step 5: Find the area of the circle** Area = π × r^2 = π × (5)^2 = π × 25 = 25π. --- **Final Answer:** 25π

3. Emma is helping her science class calculate the dimensions of a rectangular terrarium for a project. The area of the terrarium must be exactly 125 square inches. The teacher says the length should be the square root of 75 inches, and the width should be the square root of 75 inches as well. Emma's friend Liam claims the side length of the terrarium will be a rational number because 75 ends in 5. Is Liam correct? Determine whether the side length is rational or irrational, and explain your reasoning. Answer: Irrational Solution: The side length is sqrt(75) inches. Step 2: Determine if 75 is a perfect square. The perfect squares near 75 are 64 (8^2) and 81 (9^2).
   Full step-by-step solution

   Step 1: The side length is sqrt(75) inches. Step 2: Determine if 75 is a perfect square. The perfect squares near 75 are 64 (8^2) and 81 (9^2). Since 75 is not a perfect square, sqrt(75) cannot be written as an integer. Step 3: Simplify sqrt(75): sqrt(75) = sqrt(25 * 3) = sqrt(25) * sqrt(3) = 5 * sqrt(3). Step 4: sqrt(3) is irrational because it cannot be expressed as a fraction of two integers and its decimal representation does not terminate or repeat. Step 5: Multiplying a rational number (5) by an irrational number (sqrt(3)) gives an irrational number. Step 6: Therefore, sqrt(75) is irrational. Liam is incorrect; the side length is irrational.

4. Sophia is designing a circular fountain for a park. The fountain will have an area of exactly 50π square feet. She needs to determine whether the radius of the fountain is a rational or irrational number. What type of number is the radius? Explain your reasoning. Answer: irrational Solution: The area of a circle is given by A = πr². We know the area is 50π square feet, so set up the equation: πr² = 50π. Divide both sides by π: r² = 50.
   Full step-by-step solution

   Step 1: The area of a circle is given by A = πr². Step 2: We know the area is 50π square feet, so set up the equation: πr² = 50π. Step 3: Divide both sides by π: r² = 50. Step 4: Take the square root of both sides: r = √50. Step 5: Simplify √50: √50 = √(25 × 2) = √25 × √2 = 5√2. Step 6: √2 is an irrational number because it cannot be expressed as a fraction of two integers and its decimal expansion never repeats or terminates. Step 7: Multiplying a rational number (5) by an irrational number (√2) gives an irrational number. Therefore, the radius of the fountain is an irrational number.

5. Is √(27) rational or irrational? Is 22/7 rational or irrational? Answer: √(27) is irrational; 22/7 is rational Solution: Determine if √(27) is rational or irrational. √(27) = √(9 × 3) = 3√3. Since √3 is irrational (it cannot be expressed as a fraction of two integers), 3√3 is also irrational.
   Full step-by-step solution

   Step 1: Determine if √(27) is rational or irrational. √(27) = √(9 × 3) = 3√3. Since √3 is irrational (it cannot be expressed as a fraction of two integers), 3√3 is also irrational. Therefore, √(27) is irrational. Step 2: Determine if 22/7 is rational or irrational. 22/7 is a fraction of two integers (22 and 7). Any number that can be written as a fraction a/b where a and b are integers and b ≠ 0 is rational. Therefore, 22/7 is rational. The answer is: √(27) is irrational; 22/7 is rational.

6. √(196) + ∛(216) = ? Answer: 20 Solution: Evaluate the square root of 196. Since 14 × 14 = 196, √(196) = 14. Evaluate the cube root of 216.
   Full step-by-step solution

   Step 1: Evaluate the square root of 196. Since 14 × 14 = 196, √(196) = 14. Step 2: Evaluate the cube root of 216. Since 6 × 6 × 6 = 216, ∛(216) = 6. Step 3: Add the results: 14 + 6 = 20. The answer is 20.

7. Is √(361) rational or irrational? Answer: Rational Solution: Determine if 361 is a perfect square. 19 × 19 = 361, so 361 is a perfect square. Step 2: √(361) = 19.
   Full step-by-step solution

   Step 1: Determine if 361 is a perfect square. 19 × 19 = 361, so 361 is a perfect square. Step 2: √(361) = 19. Step 3: 19 is a whole number. Any whole number can be written as a fraction (19/1), so it is rational. The answer is Rational.

8. √(27) ÷ √(3) = ? Answer: 3 Solution: Use the property √(a) ÷ √(b) = √(a/b) √(27) ÷ √(3) = √(27/3) 27/3 = 9 √(9) = 3 The result is 3 The answer is 3.
   Full step-by-step solution

   Step 1: Use the property √(a) ÷ √(b) = √(a/b) √(27) ÷ √(3) = √(27/3) Step 2: Simplify the fraction inside the square root 27/3 = 9 Step 3: Evaluate the square root √(9) = 3 Step 4: The result is 3 The answer is 3.