Rational vs Irrational

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Answer Key & Explanations

Rational vs Irrational · Grade 8 · Worksheet 1

1. √(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

2. √(81) + ∛(64) - 4² = ? Answer: -3 Solution: Evaluate the square root: √(81) = 9 Evaluate the cube root: ∛(64) = 4 Evaluate the exponent: 4² = 16 Substitute the values back into the expression: 9 + 4 - 16 Perform the addition first: 9 + 4 = 13 Perform the subtraction: 13 - 16 = -3 The answer is -3.
   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: 4² = 16 Step 4: Substitute the values back into the expression: 9 + 4 - 16 Step 5: Perform the addition first: 9 + 4 = 13 Step 6: Perform the subtraction: 13 - 16 = -3 The answer is -3.

3. 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.

4. Liam is designing a rectangular garden with a length of √50 meters and a width of 3√2 meters. His friend Emma says the area must be a rational number, but Liam thinks it might be irrational. Who is correct and what is the area of the garden in simplest form? Answer: 30 square meters Solution: Write down the formula for the area of a rectangle. Area = length × width. Substitute the given values.
   Full step-by-step solution

   Step 1: Write down the formula for the area of a rectangle. Area = length × width. Step 2: Substitute the given values. Length = √50 meters Width = 3√2 meters So Area = √50 × 3√2. Step 3: Multiply the numbers. √50 × 3√2 = 3 × (√50 × √2). Step 4: Multiply the square roots using the rule √a × √b = √(a × b). √50 × √2 = √(50 × 2) = √100. Step 5: Simplify √100. √100 = 10. Step 6: Finish the multiplication. Area = 3 × 10 = 30 square meters. Step 7: Check if the result is rational or irrational. 30 is an integer, and all integers are rational numbers. Therefore, Emma is correct — the area is rational. Final answer: The area is 30 square meters.

5. Aroha is building a square patio in her backyard. The area of the patio is 45 square meters. She needs to know if the side length of the patio is a rational or irrational number so she can order the correct number of paving stones. What type of number is the side length of Aroha's patio? Answer: irrational Solution: The area of a square is given by A = s^2, where s is the side length. The area is 45 square meters, so s^2 = 45. Take the square root of both sides: s = sqrt(45).
   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: The area is 45 square meters, so s^2 = 45. Step 3: Take the square root of both sides: s = sqrt(45). Step 4: Simplify sqrt(45) by factoring: sqrt(45) = sqrt(9 * 5) = sqrt(9) * sqrt(5) = 3 * sqrt(5). Step 5: sqrt(5) is irrational because 5 is not a perfect square and its square root cannot be written as a fraction. Multiplying an irrational number (sqrt(5)) by a rational number (3) still gives an irrational number. Step 6: Therefore, the side length of Aroha's patio is an irrational number.

6. Is √(225) + ∛(512) rational or irrational? Answer: Rational Solution: Evaluate √(225). Since 15 × 15 = 225, √(225) = 15. Evaluate ∛(512).
   Full step-by-step solution

   Step 1: Evaluate √(225). Since 15 × 15 = 225, √(225) = 15. Step 2: Evaluate ∛(512). Since 8 × 8 × 8 = 512, ∛(512) = 8. Step 3: Add the results: 15 + 8 = 23. Step 4: 23 is an integer, and all integers are rational because they can be written as a fraction (23/1). The answer is Rational.

7. Noah is building a square patio in his backyard. He wants the patio to have an area of 98 square feet. He needs to order paving stones, but the supplier asks for the exact side length of the square. Noah tells his friend Ava that the side length must be an irrational number. Is Noah correct? Explain why or why not, and determine the exact side length in simplest radical form. Answer: Yes, Noah is correct; the side length is 7√2 feet, which is irrational. Solution: The area of a square is A = s², where s is the side length. We know A = 98 square feet. So s² = 98.
   Full step-by-step solution

   Step 1: The area of a square is A = s², where s is the side length. We know A = 98 square feet. So s² = 98. Step 2: To find the side length, take the square root of both sides: s = √98. Step 3: Simplify √98 by factoring: 98 = 49 × 2. So √98 = √(49 × 2) = √49 × √2 = 7√2. Step 4: Determine if 7√2 is rational or irrational. The number √2 is irrational (it cannot be expressed as a fraction of integers). Multiplying an irrational number (√2) by a rational number (7) still gives an irrational number. Step 5: Therefore, Noah is correct. The exact side length is 7√2 feet, and it is irrational because it contains √2, which is a non-repeating, non-terminating decimal. Answer: Yes, Noah is correct; the side length is 7√2 feet, which is irrational.

8. √(81) + ∛(64) - 3² = ? Answer: 4 Solution: Evaluate √(81). The square root of 81 is 9, so √(81) = 9. Evaluate ∛(64).
   Full step-by-step solution

   Step 1: Evaluate √(81). The square root of 81 is 9, so √(81) = 9. Step 2: Evaluate ∛(64). The cube root of 64 is 4, so ∛(64) = 4. Step 3: Evaluate 3². 3 squared is 9, so 3² = 9. Step 4: Substitute the values into the expression: 9 + 4 - 9. Step 5: Perform the operations from left to right: 9 + 4 = 13, then 13 - 9 = 4. The answer is 4.