Rational vs Irrational

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Answer Key & Explanations

Rational vs Irrational · Grade 8 · Worksheet 3

1. Charlotte is helping her science teacher prepare a demonstration. She needs to cut a wooden board into a square with an area of 98 square centimeters. Her teacher asks her to determine if the side length of the square is a rational or irrational number. What type of number is the side length? Explain your reasoning. Answer: Irrational Solution: The area of a square is given by A = s^2, where s is the side length. Here, A = 98, so s^2 = 98. To find s, take the square root of both sides: s = sqrt(98).
   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 = 98, so s^2 = 98. Step 2: To find s, take the square root of both sides: s = sqrt(98). Step 3: Simplify sqrt(98) by factoring: 98 = 49 * 2, so sqrt(98) = sqrt(49 * 2) = sqrt(49) * sqrt(2) = 7 * sqrt(2). Step 4: sqrt(2) is an irrational number because it cannot be expressed as a fraction of two integers. Multiplying an irrational number by a rational number (7) still gives an irrational number. Step 5: Therefore, the side length is irrational. The answer is irrational.

2. √(64) + ∛(125) - 2² = ? Answer: 9 Solution: Evaluate √(64) = 8 Evaluate ∛(125) = 5 Evaluate 2² = 4 Substitute the values: 8 + 5 - 4 Add first: 8 + 5 = 13 Subtract: 13 - 4 = 9 The answer is 9.
   Full step-by-step solution

   Step 1: Evaluate √(64) = 8 Step 2: Evaluate ∛(125) = 5 Step 3: Evaluate 2² = 4 Step 4: Substitute the values: 8 + 5 - 4 Step 5: Add first: 8 + 5 = 13 Step 6: Subtract: 13 - 4 = 9 The answer is 9.

3. Charlotte is helping her science class measure the side length of a square plot for a garden experiment. The area of the plot is 98 square feet. Charlotte needs to classify the side length as either rational or irrational. What type of number is the side length of the square plot, and why? Answer: Irrational Solution: The area of a square is given by A = s^2, where s is the side length. We know the area is 98 square feet, so s^2 = 98. To find the side length, take the square root of both sides: s = sqrt(98).
   Full step-by-step solution

   Step 1: The area of a square is given by A = s^2, where s is the side length. Step 2: We know the area is 98 square feet, so s^2 = 98. Step 3: To find the side length, take the square root of both sides: s = sqrt(98). Step 4: Simplify sqrt(98) by factoring: sqrt(98) = sqrt(49 × 2) = sqrt(49) × sqrt(2) = 7 × sqrt(2) = 7√2. Step 5: sqrt(2) is an irrational number because it cannot be expressed as a fraction of two integers, and its decimal representation (1.41421356...) never repeats or terminates. Step 6: Multiplying the rational number 7 by the irrational number sqrt(2) gives 7√2, which is also irrational. Step 7: Therefore, the side length of the square plot is an irrational number.

4. Is sqrt(121) + sqrt(81) rational or irrational? Answer: Rational Solution: Evaluate sqrt(121). Since 11 x 11 = 121, sqrt(121) = 11. Evaluate sqrt(81).
   Full step-by-step solution

   Step 1: Evaluate sqrt(121). Since 11 x 11 = 121, sqrt(121) = 11. Step 2: Evaluate sqrt(81). Since 9 x 9 = 81, sqrt(81) = 9. Step 3: Add the results: 11 + 9 = 20. Step 4: 20 is an integer, and all integers are rational numbers because they can be written as a fraction (20/1). Therefore, sqrt(121) + sqrt(81) is rational. The answer is Rational.

5. Liam is designing a rectangular garden with an area of 48 square meters. He wants the length to be the square root of a perfect square number and the width to be the square root of a non-perfect square number. If the length is √16 meters, what is the width in simplest radical form? Answer: 3√2 Solution: When working with areas of rectangles, the width can be found by dividing the area by the length.
   Full step-by-step solution

   When working with areas of rectangles, the width can be found by dividing the area by the length. In mathematics, numbers can be classified as rational (can be expressed as a fraction) or irrational (cannot be expressed as a fraction). The square roots of perfect squares are rational numbers, while the square roots of other numbers are irrational. When simplifying radicals, we look for perfect square factors.

6. Liam is designing a rectangular garden with an area of 24 square meters. He wants the length to be √18 meters and the width to be √8 meters. His friend Noah says the area won't be exactly 24 square meters because these are irrational numbers. Is Noah correct? Explain why or why not. Answer: No, Noah is not correct. The area will be exactly 24 square meters because √18 × √8 = √144 = 12, and 12 × 2 = 24. Solution: When working with square roots, multiplying two irrational numbers can sometimes produce a rational result if the product under the radical is a perfect square.
   Full step-by-step solution

   When working with square roots, multiplying two irrational numbers can sometimes produce a rational result if the product under the radical is a perfect square. For example, √2 × √8 = √16 = 4, which is rational. This occurs because the irrational parts can cancel out or combine to form rational numbers.

7. A right triangle is drawn on a coordinate plane with vertices at (0,0), (8,0), and (8,15). 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: 289π/4 Solution: Find the length of the hypotenuse using the Pythagorean theorem. The legs are 8 and 15 units. Hypotenuse = sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17 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 8 and 15 units. Hypotenuse = sqrt(8^2 + 15^2) = sqrt(64 + 225) = sqrt(289) = 17 Step 2: The hypotenuse is the diameter of the circle. Diameter = 17 Step 3: Find the radius of the circle. Radius = Diameter/2 = 17/2 Step 4: Calculate the area of the circle. Area = π × (radius)^2 = π × (17/2)^2 = π × (289/4) = 289π/4 The exact area of the circle is 289π/4.

8. Is √(225) rational or irrational? Answer: Rational Solution: Determine if 225 is a perfect square. 15 × 15 = 225, so √(225) = 15. 15 can be written as the fraction 15/1, which is a ratio of two integers.
   Full step-by-step solution

   Step 1: Determine if 225 is a perfect square. 15 × 15 = 225, so √(225) = 15. Step 2: 15 can be written as the fraction 15/1, which is a ratio of two integers. Step 3: Therefore, √(225) is rational because it equals an integer, and all integers are rational numbers. The answer is Rational.