Rational vs Irrational

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Answer Key & Explanations

Rational vs Irrational · Grade 8 · Worksheet 3

1. Is √(121) + ∛(216) rational or irrational? Answer: Rational Solution: Evaluate √(121). Since 11 × 11 = 121, √(121) = 11. Evaluate ∛(216).
   Full step-by-step solution

   Step 1: Evaluate √(121). Since 11 × 11 = 121, √(121) = 11. Step 2: Evaluate ∛(216). Since 6 × 6 × 6 = 216, ∛(216) = 6. Step 3: Add the results: 11 + 6 = 17. Step 4: 17 is an integer, and any integer can be written as a fraction (e.g., 17/1). Therefore, 17 is a rational number. The answer is Rational.

2. Mere is designing a square garden in her backyard. She draws the garden on a coordinate plane with corners at (0,0), (8,0), (8,8), and (0,8). She then draws a diagonal path from (0,0) to (8,8) and places a circular birdbath at the exact midpoint of this diagonal. The radius of the birdbath is exactly half the distance from the midpoint to the corner at (8,0). Will the area of the circular birdbath be a rational or irrational number? Explain your reasoning. Answer: Irrational Solution: Find the midpoint of the diagonal from (0,0) to (8,8). Midpoint = ((0+8)/2, (0+8)/2) = (4,4) Calculate the distance from the midpoint (4,4) to the corner (8,0).
   Full step-by-step solution

   Step 1: Find the midpoint of the diagonal from (0,0) to (8,8). Midpoint = ((0+8)/2, (0+8)/2) = (4,4) Step 2: Calculate the distance from the midpoint (4,4) to the corner (8,0). Distance = sqrt((8-4)^2 + (0-4)^2) = sqrt(4^2 + (-4)^2) = sqrt(16 + 16) = sqrt(32) = sqrt(16*2) = 4*sqrt(2) Step 3: The radius of the birdbath is half of this distance. Radius = (4*sqrt(2))/2 = 2*sqrt(2) Step 4: Calculate the area of the circular birdbath. Area = pi * (radius)^2 = pi * (2*sqrt(2))^2 = pi * 4 * 2 = 8*pi Step 5: Determine if the area is rational or irrational. 8 is rational, but pi is irrational. The product of a non-zero rational number (8) and an irrational number (pi) is always irrational. Therefore, the area of the circular birdbath (8*pi) is irrational.

3. Is sqrt(121) + sqrt(49) rational or irrational? Explain why. Answer: Rational Solution: Evaluate sqrt(121). Since 11 × 11 = 121, sqrt(121) = 11. Evaluate sqrt(49).
   Full step-by-step solution

   Step 1: Evaluate sqrt(121). Since 11 × 11 = 121, sqrt(121) = 11. Step 2: Evaluate sqrt(49). Since 7 × 7 = 49, sqrt(49) = 7. Step 3: Add the results: 11 + 7 = 18. Step 4: 18 is a whole number, which can be written as the fraction 18/1. Therefore, it is a rational number. The answer is rational.

4. Hana is building a square garden in her backyard. She wants the garden to have an area of exactly 50 square meters. Her friend Matiu says the side length of the garden will be a rational number because 50 is a whole number, but Hana thinks it might be irrational. Who is correct? Explain your reasoning by determining whether the side length is rational or irrational. Answer: Irrational Solution: The area of a square is given by A = s^2, where s is the side length. Here, A = 50, so s^2 = 50. To find the side length, take the square root of both sides: s = sqrt(50).
   Full step-by-step solution

   Step 1: The area of a square is given by A = s^2, where s is the side length. Here, A = 50, so s^2 = 50. Step 2: To find the side length, take the square root of both sides: s = sqrt(50). Step 3: Simplify sqrt(50) by factoring: sqrt(50) = sqrt(25 * 2) = sqrt(25) * sqrt(2) = 5 * sqrt(2). Step 4: sqrt(2) is an irrational number because it cannot be expressed as a fraction of two integers (it is a non-repeating, non-terminating decimal). Step 5: When you multiply a rational number (5) by an irrational number (sqrt(2)), the result is irrational. Step 6: Therefore, the side length is 5 * sqrt(2) meters, which is an irrational number. Hana is correct. The answer is irrational.

5. √(2.25) + ∛(8) = ? Answer: 3.5 Solution: √(2.25) + ∛(8) = ?
   Full step-by-step solution

   Let's solve step by step. We are given: √(2.25) + ∛(8) = ? --- **Step 1: Interpret the square root of 2.25** 2.25 can be written as a fraction: 2.25 = 225 / 100 So, √(2.25) = √(225 / 100) = √225 / √100 √225 = 15 √100 = 10 So √(2.25) = 15 / 10 = 1.5 --- **Step 2: Interpret the cube root of 8** ∛(8) means: what number cubed gives 8? We know 2 × 2 × 2 = 8, so ∛(8) = 2. --- **Step 3: Add the results** √(2.25) + ∛(8) = 1.5 + 2 = 3.5 --- **Final Answer:** 3.5

6. √(144) + ∛(125) - 6² = ? Answer: -19 Solution: Evaluate the square root: √(144) = 12 Evaluate the cube root: ∛(125) = 5 Evaluate the exponent: 6² = 36 Substitute the values back into the expression: 12 + 5 - 36 Perform the addition first: 12 + 5 = 17 Perform the subtraction: 17 - 36 = -19 The answer is -19.
   Full step-by-step solution

   Step 1: Evaluate the square root: √(144) = 12 Step 2: Evaluate the cube root: ∛(125) = 5 Step 3: Evaluate the exponent: 6² = 36 Step 4: Substitute the values back into the expression: 12 + 5 - 36 Step 5: Perform the addition first: 12 + 5 = 17 Step 6: Perform the subtraction: 17 - 36 = -19 The answer is -19.

7. √(81) + ∛(64) - 2³ = ? Answer: 5 Solution: Evaluate the square root: √(81) = 9 Evaluate the cube root: ∛(64) = 4 Evaluate the exponent: 2³ = 8 Substitute the values: 9 + 4 - 8 Perform the addition: 9 + 4 = 13 Perform the subtraction: 13 - 8 = 5 The answer is 5.
   Full step-by-step solution

   Step 1: Evaluate the square root: √(81) = 9 Step 2: Evaluate the cube root: ∛(64) = 4 Step 3: Evaluate the exponent: 2³ = 8 Step 4: Substitute the values: 9 + 4 - 8 Step 5: Perform the addition: 9 + 4 = 13 Step 6: Perform the subtraction: 13 - 8 = 5 The answer is 5.

8. A right triangle is drawn on a coordinate plane with vertices at (0,0), (5,0), and (5,12). A circle is drawn such that its diameter is equal to the length of the hypotenuse of this triangle. What is the exact area of the circle? Express your answer in terms of π. Answer: 169π/4 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs are 5 units and 12 units. Hypotenuse = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 units The hypotenuse is the diameter of the circle.
   Full step-by-step solution

   Step 1: Find the length of the hypotenuse using the Pythagorean theorem. The legs are 5 units and 12 units. Hypotenuse = sqrt(5^2 + 12^2) = sqrt(25 + 144) = sqrt(169) = 13 units Step 2: The hypotenuse is the diameter of the circle. Diameter = 13 units Step 3: Find the radius of the circle. Radius = Diameter/2 = 13/2 units Step 4: Calculate the area of the circle. Area = π × (radius)^2 = π × (13/2)^2 = π × (169/4) = 169π/4 The exact area of the circle is 169π/4.