Is √(2/9) rational or irrational? — Rational — it equals √2/3, and √2/3 … wait, no. √(2/9) = √2/3 which is irrational.√(2/9) = √2/√9 = √2/3. Since √2 is irrational and 3 is rational, the quotient is irrational.
Which of these is a rational number between √2 and √3? — 1.5.√2 ≈ 1.414 and √3 ≈ 1.732. The value 1.5 is between them and is clearly rational (= 3/2). √2.5 is irrational.
A student writes: "The decimal 0.123456789101112… is rational because it contains only digits." Are they correct? — No — the decimal is non-repeating and non-terminating, so it is irrational.The Champernowne-like decimal 0.1234… never enters a repeating cycle, making it irrational despite having a visible construction pattern.
The number −√2 is best classified as: — Irrational.The negative sign does not change irrationality. −√2 is still non-terminating and non-repeating, hence irrational.
Between consecutive integers 4 and 5, which irrational number lies closest to 5? — √24.√24 ≈ 4.899, √17 ≈ 4.123, √18 ≈ 4.243, √20 ≈ 4.472. √24 is closest to 5 among these choices.
If p is rational and q is irrational, which expression is always irrational (assume p ≠ 0)? — p + q.A nonzero rational plus an irrational is always irrational. p × p = p² is rational; q × 0 = 0 is rational; p − p = 0 is rational.
What is the relationship between √n and √(n+1) for large positive integers n? — √(n+1) > √n, but the difference gets smaller as n grows.Square roots are increasing but concave — the gaps between consecutive square roots shrink as n grows larger.
If 4.4² = 19.36 and 4.5² = 20.25, what is the best two-decimal approximation of √20? — 4.47.4.47² = 19.9809 ≈ 20. Checking: 4.47 is between 4.44 (= 19.71) and 4.50 (= 20.25), and 4.47 is closest.
Without a calculator, order these from least to greatest: √5, 2.3, √6, 2.5. — √5 < 2.3 < √6 < 2.5.√5 ≈ 2.236, √6 ≈ 2.449. Order: √5 ≈ 2.236 < 2.3 < √6 ≈ 2.449 < 2.5.
Which of the following is a correct strategy to estimate √n to the hundredths? — Squeeze between tenths: find a and a.1 such that a² < n < (a.1)², then try a.05.Systematic squeezing — narrowing the interval by testing midpoints — is the correct iterative approximation strategy.
Using interval narrowing, a student finds 3.16² = 9.9856 and 3.17² = 10.0489. What is the best one-decimal approximation of √10? — 3.2.√10 is between 3.16 and 3.17, so to the nearest tenth it rounds to 3.2.
Which list correctly orders √2, √3, 1.5, and 1.8 from least to greatest? — √2 < 1.5 < √3 < 1.8.√2 ≈ 1.414, 1.5, √3 ≈ 1.732, 1.8. So √2 < 1.5 < √3 < 1.8.
Which statement about the real number system is FALSE? — Real numbers are closed under division (any two reals can be divided).Division by zero is undefined. Since 0 is a real number and 5 ÷ 0 is undefined, the reals are NOT closed under division.
If a is irrational and b is rational (b ≠ 0), is a/b rational or irrational? — Irrational.Dividing an irrational by a nonzero rational always gives an irrational. E.g., √2/3 is irrational.
Which of the following does NOT follow from the field properties of real numbers? — The sum of two primes is prime.2 + 3 = 5 (both prime, prime result) but 2 + 7 = 9 (not prime). The other three are guaranteed field axioms.
Between √2 and √3, how many rational numbers exist? — Infinitely many.By the density property, infinitely many rational numbers exist between any two distinct reals.
Which property is used in every step of solving the equation 3x + 6 = 15? — Properties of equality combined with inverse and identity properties.Solving equations uses subtraction/addition (inverse) and division/multiplication (inverse) properties together with equality properties.