Grade 9 Number & Quantity Practice

Question 1 of 27Score 0/0Medium

Question 1 of 27: Which is the correct hierarchy from smallest to largest set?

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Keep going: Read the lesson: The Real Number System
Answer key for parents & teachers (27 questions)
  1. Which is the correct hierarchy from smallest to largest set? โ€” Natural โŠ‚ Whole โŠ‚ Integers โŠ‚ Rational. Natural (1,2,3โ€ฆ) โŠ‚ Whole (0,1,2โ€ฆ) โŠ‚ Integers (โ€ฆโˆ’1,0,1โ€ฆ) โŠ‚ Rational โŠ‚ Real.
  2. What is the sum 2 + โˆš3 classified as? โ€” Irrational. Rational + irrational = irrational. Since โˆš3 is irrational, 2 + โˆš3 is also irrational.
  3. Which statement is ALWAYS true? โ€” All natural numbers are integers. Natural โŠ‚ Integer, so every natural number IS an integer. The other statements are false.
  4. Convert 0.777... to a fraction. โ€” 7/9. Let x = 0.777... Then 10x = 7.777... Subtract: 9x = 7 โ†’ x = 7/9.
  5. โˆš2 ร— โˆš2 is equal to: โ€” โˆš4 = 2 (rational). โˆš2 ร— โˆš2 = (โˆš2)ยฒ = 2. The product of an irrational number with itself can be rational!
  6. Between which two consecutive integers does โˆš50 lie? โ€” 7 and 8. 7ยฒ = 49 and 8ยฒ = 64. Since 49 < 50 < 64, we have 7 < โˆš50 < 8.
  7. The real number line contains: โ€” All rational and irrational numbers together. The real number line represents ALL real numbers โ€” both rational (fractions, integers) and irrational (โˆš2, ฯ€, e) โ€” with no gaps.
  8. Evaluate 16^(3/4). โ€” 8. 16^(3/4) = (โดโˆš16)ยณ = 2ยณ = 8.
  9. What is 8^(โˆ’1/3)? โ€” 1/2. 8^(โˆ’1/3) = 1/8^(1/3) = 1/โˆ›8 = 1/2.
  10. Simplify 32^(2/5). โ€” 4. 32^(2/5) = (โตโˆš32)ยฒ = 2ยฒ = 4.
  11. Simplify x^(4/6) in lowest terms. โ€” x^(2/3). Reduce the fraction: 4/6 = 2/3. So x^(4/6) = x^(2/3).
  12. Which of these is equivalent to (โˆ›x)โด? โ€” x^(4/3). (โˆ›x)โด = (x^(1/3))โด = x^(4/3). Multiply exponents: 1/3 ร— 4 = 4/3.
  13. Evaluate (โˆ’8)^(1/3). โ€” โˆ’2. Cube roots of negative numbers are real and negative. (โˆ’8)^(1/3) = โˆ›(โˆ’8) = โˆ’2, since (โˆ’2)ยณ = โˆ’8.
  14. Rewrite a^(3/4) ยท a^(1/4) using a single exponent. โ€” a^(1). Product rule: add exponents. 3/4 + 1/4 = 4/4 = 1. So the result is aยน = a.
  15. Simplify (x^(1/2))^6. โ€” xยณ. Power rule: multiply exponents. (1/2) ร— 6 = 3. Answer: xยณ.
  16. What is 0^(1/2)? โ€” 0. 0^(1/2) = โˆš0 = 0. Zero raised to any positive power is 0.
  17. Why is (โˆ’4)^(1/2) not a real number? โ€” Because โˆ’4 is negative and even roots of negatives are not real. Even-index roots of negative numbers are not real (they are complex). โˆš(โˆ’4) = 2i, which is imaginary, not real.
  18. What is i^12? โ€” 1. 12 รท 4 = 3 remainder 0. Remainder 0 โ†’ i^0 = 1. So i^12 = 1.
  19. Simplify โˆš(โˆ’72). โ€” 6iโˆš2. โˆš(โˆ’72) = โˆš(36 ยท 2 ยท (โˆ’1)) = 6โˆš2 ยท i = 6iโˆš2.
  20. What is i^31? โ€” โˆ’i. 31 รท 4 = 7 remainder 3. Remainder 3 โ†’ โˆ’i. So i^31 = โˆ’i.
  21. Which of the following is a pure imaginary number? โ€” 7i. A pure imaginary number has the form bi (b โ‰  0, no real part). 7i is pure imaginary.
  22. Simplify i^100. โ€” 1. 100 รท 4 = 25 remainder 0. Remainder 0 โ†’ 1. So i^100 = 1.
  23. What is (2i)ยฒ? โ€” โˆ’4. (2i)ยฒ = 4 ยท iยฒ = 4 ยท (โˆ’1) = โˆ’4.
  24. The imaginary unit i was invented to solve which type of problem? โ€” Square roots of negative numbers. i was introduced to handle โˆš(negative number), specifically to give meaning to โˆš(โˆ’1).
  25. Simplify: iโต. โ€” i. 5 รท 4 = 1 remainder 1. Remainder 1 โ†’ i. So iโต = i. (Alternatively, iโต = iโด ยท i = 1 ยท i = i.)
  26. What is the period of the powers of i? โ€” 4. The powers of i cycle every 4: i, โˆ’1, โˆ’i, 1, i, โˆ’1, โˆ’i, 1, โ€ฆ The period is 4.
  27. The complex conjugate of 4 + 3i is: โ€” 4 โˆ’ 3i. The complex conjugate of a + bi is a โˆ’ bi. So the conjugate of 4 + 3i is 4 โˆ’ 3i.