Is the product of two irrational numbers always irrational? — No — √2 × √2 = 2 is rational.√2 × √2 = 2, which is rational. So the product of two irrationals is not always irrational.
Which set is CLOSED under subtraction (subtracting always stays in the set)? — Integers.Integers are closed under subtraction: integer − integer = integer. Natural and whole numbers fail (e.g., 3−5 = −2 is not natural/whole). Irrationals fail (√2 − √2 = 0, which is rational).
A student claims that between any two rational numbers there is always an irrational number. Is this true? — True — irrationals are dense in the reals too.Between any two rationals there are infinitely many irrationals (and vice versa). Both rationals and irrationals are dense in the reals.
If a is rational and b is irrational, what can you say about a/b? — Always irrational (unless a = 0).If a ≠ 0, then a/b is irrational. If a = 0, then 0/b = 0, which is rational. So: irrational unless a = 0.
Classify: the square root of a perfect square integer is always a(n): — Integer.Perfect squares (1, 4, 9, 16, …) have integer square roots (1, 2, 3, 4, …). Their roots are integers — the most specific classification.
Which is larger: √2 + √3 or √5? — √2 + √3.√2 ≈ 1.414, √3 ≈ 1.732, sum ≈ 3.146. √5 ≈ 2.236. So √2 + √3 > √5. Note: (√2 + √3)² = 5 + 2√6 > 5 = (√5)².
Is 0.101001000100001... (a non-repeating pattern) rational or irrational? — Irrational, because it neither terminates nor repeats.For a decimal to be rational it must terminate OR repeat (a fixed block cycles). This decimal has a pattern but is NOT repeating — the 1s occur after increasing gaps. It is irrational.
How many irrational numbers are there between 1 and 2? — Infinitely many.There are infinitely many (in fact uncountably infinitely many) irrationals between any two real numbers.
Which operation on two irrational numbers can yield a rational result? — Both addition and multiplication.Addition: √2 + (−√2) = 0 (rational). Multiplication: √2 × √2 = 2 (rational). Both operations can yield rational results.
The number e ≈ 2.71828... is: — Irrational and transcendental.e (Euler's number) is irrational (non-terminating, non-repeating) and transcendental (not a root of any polynomial with integer coefficients).
If x is irrational, is x² always irrational? — No — (√3)² = 3 is rational.(√3)² = 3, which is rational. Squaring an irrational number can produce a rational result.
Solve for x: x^(3/2) = 27. — x = 9.x^(3/2) = 27. Raise both sides to the 2/3 power: x = 27^(2/3) = (∛27)² = 3² = 9.
If 2^x = 8, express x as a rational exponent relationship. — x = log₂8 = 3.2^x = 8 = 2³ → x = 3. This is equivalent to log₂8 = 3. Rational exponents and logarithms are closely connected.
Evaluate (−27)^(2/3). — 9.(−27)^(2/3) = ((−27)^(1/3))² = (−3)² = 9. The cube root of −27 is −3, and (−3)² = 9.
Which is larger: 4^(3/2) or 8^(2/3)? — 4^(3/2) = 8 is larger.4^(3/2) = (√4)³ = 2³ = 8. 8^(2/3) = (∛8)² = 2² = 4. So 8 > 4, meaning 4^(3/2) is larger.
Solve: x^(2/3) = 4. — x = ±8.Raise both sides to 3/2: x = ±4^(3/2) = ±(√4)³ = ±8. Since x^(2/3) is defined for negative x (cube root, then square), both ±8 are solutions.
What type of number is a + bi where a ≠ 0 and b ≠ 0? — Complex (with real and imaginary parts).A number of the form a + bi with both a ≠ 0 and b ≠ 0 is a complex number that is neither purely real nor purely imaginary.
Is every real number also a complex number? — Yes — a real number a = a + 0i is complex.Complex numbers a + bi include all reals (when b = 0). Real numbers are a special case of complex numbers.