What type of number is 0.12112111211112โฆ? โ Irrational โ pattern never exactly repeats.Although it has a pattern, the block length keeps growing so it never forms a true repeating decimal. It is irrational.
Is the product of two irrational numbers always irrational? โ No โ โ2 ร โ2 = 2 is rational.โ2 ร โ2 = 2, a rational number. This is a counterexample showing the product need not be irrational.
Which number has a terminating decimal expansion? โ 3/8.3/8 = 0.375 (terminates). The denominators 3, 7, and 9 have prime factors other than 2 and 5, giving repeating decimals.
Is the sum of a rational number and an irrational number rational or irrational? โ Always irrational.Adding a nonzero rational to an irrational always yields an irrational. E.g., 1 + โ2 is irrational.
A classmate says "All decimals are irrational." What is the best counterexample? โ 0.5 = 1/2 is a terminating decimal and is rational.0.5 is both a decimal and a rational number, directly disproving the claim.
If โn is irrational, what can you conclude about n? โ n is not a perfect square.A square root is irrational if and only if the radicand is not a perfect square. Odd/prime/large are irrelevant โ โ4 = 2 is rational.
Classify: the cube root ยณโ8. โ Rational โ ยณโ8 = 2.ยณโ8 = 2 because 2ยณ = 8. It equals an integer, so it is rational.
Which decimal represents an irrational number? โ 3.14159265โฆ (non-repeating, non-terminating).Only non-terminating, non-repeating decimals are irrational. The other three are rational.
The set of irrational numbers is a subset of which larger set? โ Real numbers.Real numbers = rational โช irrational. Irrational numbers are real but are not rational, integer, or whole.
Which property distinguishes the rational numbers from the irrational numbers? โ Rational numbers can be expressed as a ratio of integers; irrationals cannot.The defining property is the ability (or inability) to write the number as p/q with integer p, q and q โ 0.
Evaluate: Is (โ3)ยฒ rational or irrational? โ Rational โ (โ3)ยฒ = 3.(โ3)ยฒ = 3, a positive integer. Squaring the square root cancels the radical.
To place โ30 on a number line, you first determine it is between 5 and 6. Which test narrows it further? โ Try 5.5: 5.5ยฒ = 30.25, so โ30 < 5.5.5.5ยฒ = 30.25 > 30, so โ30 must be slightly less than 5.5. This narrows the interval to (5, 5.5).
Which of these is the best approximation for โ52? โ 7.2.7.2ยฒ = 51.84 โ 52, which is much closer than the other choices. (7.21ยฒ โ 51.98)
A number line shows integers 4, 5, 6. Where should โ26 be placed? โ Between 5 and 6, closer to 5.5ยฒ = 25, so โ26 is just above 5. 5.1ยฒ = 26.01, confirming โ26 โ 5.1, which is closer to 5 than to 6.
Which value is closest to โ80? โ 8.9.8.9ยฒ = 79.21 and 9.0ยฒ = 81. โ80 โ 8.944, so 8.9 is the closest listed value.
If you know 6ยฒ = 36 and 7ยฒ = 49, where does โ40 fall on the number line? โ Between 6 and 7, closer to 6.6.3ยฒ = 39.69 and 6.4ยฒ = 40.96, so โ40 โ 6.32, which is closer to 6 than to 7.
A student says โ18 โ 4.2. Is this a good approximation? โ Yes โ 4.2ยฒ = 17.64, which is close to 18.4.2ยฒ = 17.64 and 4.24ยฒ โ 17.98. The approximation 4.2 is reasonable; the true value is about 4.243.
โ150 is closest to which value? โ 12.2.12ยฒ = 144 and 13ยฒ = 169, so โ150 is between 12 and 13. 12.2ยฒ = 148.84 and 12.3ยฒ = 151.29, so โ150 โ 12.2.
If a square has area 50 mยฒ, what is the approximate side length? โ 7.1 m.Side = โ50 โ 7.07 m. Since 7ยฒ = 49 and 7.1ยฒ = 50.41, the side is approximately 7.1 m.
Which number is between โ8 and โ15? โ โ12.โ8 โ 2.83 and โ15 โ 3.87. โ12 โ 3.46, which falls in that range. โ2 < โ8, โ16 = 4 > โ15, โ7 < โ8.
A tile is square with area 72 inยฒ. Between which two whole-number inch lengths is the side? โ 8 in and 9 in.8ยฒ = 64 < 72 < 81 = 9ยฒ, so the side โ72 is between 8 and 9 inches.
A student estimates โ35 โ 6. Without a calculator, is this estimate too high, too low, or exact? โ Too high โ 6ยฒ = 36 > 35.6ยฒ = 36 > 35, meaning 6 is slightly above โ35 โ 5.916. The estimate is too high.
The density property of real numbers states that: โ Between any two real numbers there is always another real number.Density means you can always find a real number between any two given real numbers โ there are no "jumps."
Which of the following is an example of closure under addition for real numbers? โ 3.7 + โ2 is a real number.Closure under addition means the sum of any two real numbers is also a real number.
Is the set of irrational numbers closed under addition? โ No โ โ2 + (โโ2) = 0, which is rational.โ2 + (โโ2) = 0, a rational number. This counterexample shows irrationals are not closed under addition.
Which property justifies: a ร (b ร c) = (a ร b) ร c? โ Associative property of multiplication.Regrouping factors โ without changing their order โ is the associative property of multiplication.
Using the distributive property, simplify: 7(3 + x) โ 7(3). โ 7x.7(3 + x) = 21 + 7x, then 21 + 7x โ 21 = 7x. Distributing then combining like terms.
The real number line is "complete." This means: โ Every point on the line corresponds to a real number and vice versa.Completeness means there is a one-to-one correspondence between real numbers and points on the number line โ no gaps exist.
Which property allows you to rewrite 2x + 2y as 2(x + y)? โ Distributive property (in reverse โ factoring).2x + 2y = 2(x + y) is the distributive property applied in reverse (factoring out the common factor 2).
Justify: โ3(x โ 4) = โ3x + 12. โ Distributive property: โ3 ร x = โ3x and โ3 ร (โ4) = +12.Distribute โ3 over (x โ 4): (โ3)(x) = โ3x and (โ3)(โ4) = 12.
The property that ensures the product of any real number and 0 is 0 is called the: โ Zero product property (multiplication property of zero).The multiplication property of zero states a ร 0 = 0 for any real a.
Which pair of numbers are additive inverses? โ โ5 and โโ5.Additive inverses sum to 0: โ5 + (โโ5) = 0. The others do not sum to zero.
Simplify using properties: 4 ยท (1/4) ยท x. โ x.4 ร (1/4) = 1 by the multiplicative inverse property, and 1 ร x = x by the identity property.