Approximate Irrational Numbers - 8th Grade Math

1. What inequality places $\sqrt{30}$ between two consecutive integers?

   $5<\sqrt{30}<6$. Since $5^2=25<30<36=6^2$.

2. Estimate $sqrt{3}$ to the nearest tenth by comparing $1.7^2=2.89$ and $1.8^2=3.24$.

   $\sqrt{3} \approx 1.7$. Since $2.89$ is closer to $3$ than $3.24$, round to $1.7$.

3. Which decimal form indicates an irrational number: terminating, repeating, or nonterminating nonrepeating?

   Nonterminating, nonrepeating. Rational decimals either terminate or repeat; irrationals do neither.

4. What is the definition of an irrational number?

   A number that cannot be written as $\frac{a}{b}$ for integers $a,b$ with $b\ne 0$. These numbers have no ratio representation with integers.

5. Estimate $sqrt{50}$ to the nearest tenth using $7.1^2=50.41$ and $7.0^2=49$.

   $\sqrt{50} \approx 7.1$. Since $50.41$ is very close to $50$, round down to $7.1$.

6. Estimate $pi^2$ using $pi \approx 3.14$.

   $\pi^2 \approx 9.86$. Square the approximation: $3.14^2 = 9.8596 approx 9.86$.

7. Which number is larger: $sqrt{18}$ or $sqrt{20}$?

   $\sqrt{20}$. Larger number under the radical gives larger square root.

8. Which is larger: $\sqrt{7}$ or $2.6$?

   $\sqrt{7}$. $\sqrt{7} \approx 2.646$, which is greater than $2.6$.

9. What is a reasonable estimate for $\sqrt{20}$ to the nearest tenth?

   $4.5$. $\sqrt{20} \approx 4.472$, which rounds to $4.5$.

10. What is $\pi^2$ to the nearest tenth using $\pi\approx 3.14$?

    $9.9$. $(3.14)^2=9.8596\approx 9.9$.

11. What is the definition of an irrational number in terms of its decimal form?

    A number with a nonterminating, nonrepeating decimal. Irrational decimals go on forever without repeating patterns.

12. What is $\sqrt{2}$ rounded to the nearest tenth?

    $1.4$. $\sqrt{2} \approx 1.414$, which rounds to $1.4$.

13. Which is closest to $\sqrt{3}$: $1.6$, $1.7$, or $1.8$?

    $1.7$. $\sqrt{3} \approx 1.732$, closest to $1.7$.

14. Which decimal is a closer approximation to $\sqrt{3}$: $1.7$ or $1.8$?

    $1.7$. Since $1.7^2=2.89$ is closer to $3$ than $1.8^2=3.24$.

15. What is the best integer estimate for $\sqrt{90}$?

    $9$. Since $9^2=81<90<100=10^2$, and $90$ is closer to $81$.

16. Which is closer to $sqrt{2}$: $1.41$ or $1.42$?

    $1.41$. Since $1.41^2=1.9881$ is closer to $2$ than $1.42^2=2.0164$.

17. Which inequality correctly orders the numbers $1.7$, $sqrt{3}$, and $1.8$?

    $1.7<\sqrt{3}<1.8$. Since $1.7^2=2.89<3<3.24=1.8^2$.

18. What is the definition of a rational number?

    A number expressible as $\frac{a}{b}$ where $a,b$ are integers and $b\ne 0$. The ratio form distinguishes rational from irrational numbers.

19. Estimate $2pi$ using $pi \approx 3.14$.

    $2\pi \approx 6.28$. Multiply: $2 \times 3.14 = 6.28$.

20. What is $\sqrt{2}$ rounded to the nearest hundredth?

    $1.41$. $\sqrt{2} \approx 1.414$, which rounds to $1.41$.

21. What is a rational approximation of an irrational number?

    A nearby rational number, often a truncated or rounded decimal. Used to estimate and compare irrational values.

22. What inequality shows $\sqrt{2}$ is between $1.4$ and $1.5$?

    $1.4<\sqrt{2}<1.5$. Since $1.4^2=1.96<2<2.25=1.5^2$.

23. Which is larger: $\sqrt{5}$ or $2.2$?

    $\sqrt{5}$. $\sqrt{5} \approx 2.236$, which is greater than $2.2$.

24. What is a quick estimate for $\sqrt{2}+\sqrt{3}$ to the nearest tenth?

    $3.1$. $\sqrt{2}\approx 1.41$ and $\sqrt{3}\approx 1.73$, sum is $3.14\approx 3.1$.

25. What is an irrational number (in terms of its decimal form)?

    A number with a decimal that is nonterminating and nonrepeating. Cannot be expressed as a fraction of integers.

26. What is $\sqrt{18}$ to the nearest tenth?

    $4.2$. $\sqrt{18}\approx 4.24$, which rounds to $4.2$.

27. Which is larger: $\sqrt{50}$ or $7$?

    $\sqrt{50}$. Since $7^2=49<50$, we have $7<\sqrt{50}$.

28. Which is larger: $\sqrt{17}$ or $4.2$, using quick squaring?

    $4.2$. Since $4.2^2=17.64>17$, so $\sqrt{17}<4.2$.

29. What inequality compares $\sqrt{3}$ and $\sqrt{5}$ correctly?

    $\sqrt{3}<\sqrt{5}$. Since $3<5$, their square roots maintain the same order.

30. Which inequality is true: $6<\sqrt{40}<7$ or $7<\sqrt{40}<8$?

    $6<\sqrt{40}<7$. Since $6^2=36$ and $7^2=49$, and $40$ is between them.

31. Identify the best first integer bounds for $sqrt{2}$: which inequality is correct?

    $1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $1<2<4$.

32. Estimate $\pi^2$ using $\pi\approx 3.14$ (round to the nearest hundredth).

    $9.86$. Calculate $3.14 \times 3.14 = 9.8596 \approx 9.86$.

33. What is $\sqrt{0.5}$ to the nearest hundredth?

    $0.71$. $\sqrt{0.5}=\frac{1}{\sqrt{2}}\approx 0.707$, rounds to $0.71$.

34. Which is closer to $sqrt{5}$: $2.23$ or $2.24$?

    $2.24$. Since $2.24^2=5.0176$ is closer to $5$ than $2.23^2=4.9729$.

35. What is the approximate value of $\sqrt{49}$, and is it rational or irrational?

    $7$, rational. $\sqrt{49}=7$ exactly, making it rational.

36. Which inequality is true: $1<\sqrt{2}<2$ or $2<\sqrt{2}<3$?

    $1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $2$ is between them.

37. What is a common rational approximation for $\pi$ used in estimation?

    $3.14$. Standard approximation for calculations.

38. Which is larger: $\pi$ or $3.14$?

    $\pi$. $\pi \approx 3.14159$, which is greater than $3.14$.

39. Estimate $sqrt{50}$ using the nearest perfect squares.

    $7<\sqrt{50}<8$. Since $7^2=49$ and $8^2=64$, and $49<50<64$.

40. What inequality shows $\sqrt{2}$ is between $1$ and $2$ using perfect squares?

    $1^2<2<2^2$, so $1<\sqrt{2}<2$. Since $\sqrt{2}$ squared equals $2$, it lies between the square roots of $1$ and $4$.

41. What inequality places $\sqrt{5}$ between two consecutive integers?

    $2<\sqrt{5}<3$. Since $2^2=4<5<9=3^2$, take square roots.

42. What is the approximate value of $sqrt{2}$ to the nearest hundredth?

    $\sqrt{2} \approx 1.41$. Standard approximation memorized for common calculations.

43. Identify the correct placement: is $sqrt{2}$ closer to $1$ or to $2$ on a number line?

    Closer to $1$. Since $1.41$ is less than $1.5$, it's closer to $1$.

44. What inequality shows $sqrt{2}$ is between $1.4$ and $1.5$ using squares?

    $1.4^2<\sqrt{2}^2<1.5^2$. Since $1.4^2=1.96$ and $1.5^2=2.25$, and $1.96<2<2.25$.

45. On a number line, between which integers does $\sqrt{30}$ lie?

    Between $5$ and $6$. Since $5^2=25$ and $6^2=36$, and $30$ is between them.

46. What is $\sqrt{2}+\sqrt{2}$ simplified?

    $2\sqrt{2}$. Factor out the common $\sqrt{2}$: $\sqrt{2}(1+1)=2\sqrt{2}$.

47. What is a reasonable estimate for $\pi^2$ to the nearest whole number?

    $10$. $\pi^2 \approx 9.87$, which rounds to $10$.

48. What is $\sqrt{50}$ to the nearest tenth?

    $7.1$. $\sqrt{50}\approx 7.07$, which rounds to $7.1$.

49. Which number line location is closest to $\sqrt{20}$: $4.4$ or $4.5$?

    $4.5$. Since $4.5^2=20.25$ is closer to $20$ than $4.4^2=19.36$.

50. Which is larger: $\sqrt{10}$ or $3.1$?

    $\sqrt{10}$. $\sqrt{10} \approx 3.162$, which is greater than $3.1$.

51. Identify the better estimate for $\sqrt{2}$: $1.4$ or $1.5$?

    $1.4$. Since $1.4^2=1.96$ is closer to $2$ than $1.5^2=2.25$.

52. Which inequality is true: $1.41<\sqrt{2}<1.42$ or $1.42<\sqrt{2}<1.43$?

    $1.41<\sqrt{2}<1.42$. Since $1.41^2=1.9881$ and $1.42^2=2.0164$, and $2$ is between them.

53. Which point is closest to $\sqrt{7}$ on a number line: $2.6$, $2.7$, or $2.8$?

    $2.6$. $\sqrt{7}\approx 2.646$, which is closest to $2.6$.

54. What inequality places $\sqrt{50}$ between two consecutive integers?

    $7<\sqrt{50}<8$. Since $7^2=49<50<64=8^2$.

55. Which inequality is true: $1.4<\sqrt{2}<1.5$ or $1.5<\sqrt{2}<1.6$?

    $1.4<\sqrt{2}<1.5$. Since $1.4^2=1.96$ and $1.5^2=2.25$, and $2$ is between them.

56. What is $\pi$ rounded to the nearest hundredth?

    $3.14$. $\pi\approx 3.14159...$, so it rounds to $3.14$.

57. What is the definition of a rational number in terms of fractions?

    A number that can be written as $\frac{a}{b}$ with integers $a,b$ and $b \neq 0$. Any fraction of integers (with non-zero denominator) is rational.

58. Estimate $2\pi$ using $\pi\approx 3.14$.

    $6.28$. Multiply: $2 \times 3.14 = 6.28$.

59. What is $(\sqrt{2})^2$ exactly?

    $2$. Squaring a square root gives the original number.

60. Which number is larger: $sqrt{3}$ or $sqrt{5}$?

    $\sqrt{5}$. Since $5>3$, we have $sqrt{5}>sqrt{3}$.

61. What is a reasonable estimate for $\sqrt{0.5}$ to the nearest hundredth?

    $0.71$. $\sqrt{0.5} \approx 0.707$, which rounds to $0.71$.

62. Which inequality correctly shows that $\sqrt{2}$ is between two consecutive integers?

    $1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $1<2<4$.

63. What is a common rational approximation for $pi$ to the nearest hundredth?

    $3.14$. Standard approximation of $pi$ rounded to hundredths.

64. What inequality places $\sqrt{2}$ between hundredths: $1.41$ or $1.42$?

    $1.41<\sqrt{2}<1.42$. Since $1.41^2=1.9881<2<2.0164=1.42^2$.

65. Which decimal is a closer approximation to $\sqrt{6}$: $2.4$ or $2.5$?

    $2.4$. Since $2.4^2=5.76$ is closer to $6$ than $2.5^2=6.25$.

66. Which is larger: $\sqrt{10}$ or $3.1$, using quick squaring?

    $\sqrt{10}$. Since $3.1^2=9.61<10$, so $3.1<\sqrt{10}$.

67. Which is larger: $\sqrt{8}$ or $2.8$, using quick squaring?

    $\sqrt{8}$. Since $2.8^2=7.84<8$, so $2.8<\sqrt{8}$.

68. What is an irrational number?

    A number that cannot be written as $ $\frac{a}{b}$ with integers $a,b$ and $b\ne 0$. Cannot be expressed as a ratio of integers.

69. Which is larger: $\sqrt{5}$ or $2.2$, using quick squaring?

    $\sqrt{5}$. Since $2.2^2=4.84<5$, so $2.2<\sqrt{5}$.

70. What does it mean to approximate an irrational number with a rational number?

    Use a fraction or terminating/repeating decimal close to the irrational value. Replace the irrational with a nearby rational value.

71. Which statement is always true about squaring positive numbers: if $a<b$, then what about $a^2$ and $b^2$?

    If $0<a<b$, then $a^2<b^2$. Squaring preserves inequality for positive numbers.

72. Which is larger: $\pi$ or $\sqrt{10}$?

    $\sqrt{10}$. Since $\pi \approx 3.14$ and $\sqrt{10} \approx 3.16$.

73. What is $\sqrt{2}$ rounded to the nearest hundredth?

    $1.41$. $\sqrt{2}\approx 1.414$, which rounds to $1.41$.

74. Identify the best first integer bounds for $\sqrt{10}$: which inequality is correct?

    $3<\sqrt{10}<4$. Since $3^2=9$ and $4^2=16$, and $9<10<16$.

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