Approximate Values of Irrational Numbers Quick Check
3 of 53 of 5 Items
Question
Using rational approximations, what statement is true?(1 point)
Responses
49−−√>7
start root 49 end root greater than 7
48−−√>36−−√
start root 48 end root greater than start root 36 end root
49−−√<7
start root 49 end root less than 7
48−−√<36−−√
start root 48 end root less than start root 36 end root
Using rational approximations, what statement is true?(1 point)
Responses
12 −−−√<π
start root 12 end root less than pi
16 −−−√<4
start root 16 end root less than 4
16−−√>4
start root 16 end root greater than 4
12 −−−√ >π
start root 48 end root less than start root 36 end root
To approximate the values of irrational numbers, you can use rational approximations. Rational approximations are fractions that are close to the value of the irrational number.
Let's analyze the statements one by one:
1. 49√ > 7: To determine if this statement is true, we need to find rational approximations for both numbers. The square root of 49 is 7, which is equal to 7/1. Comparing it to 7, which is equal to 7/1, we can see that 7/1 is greater than 7/1. Therefore, the statement 49√ > 7 is not true.
2. 48√ > 36√: Again, we need to find rational approximations for both numbers. The square root of 48 is approximately 6.928203, which is close to the fraction 97/14. The square root of 36 is 6, which is equal to 6/1. Comparing the two rational approximations, we can see that 97/14 is greater than 6/1. Therefore, the statement 48√ > 36√ is true.
3. 49√ < 7: By finding the rational approximations for both numbers, we can determine if this statement is true. The square root of 49 is 7, which is equal to 7/1. Comparing it to 7/1, we can see that it is equal to 7/1. Therefore, the statement 49√ < 7 is not true.
4. 48√ < 36√: In order to determine if this statement is true, we need the rational approximations for both numbers. The square root of 48 is approximately 6.928203, which is close to the fraction 97/14. The square root of 36 is 6, which is equal to 6/1. Comparing the two rational approximations, we can see that 97/14 is greater than 6/1. Therefore, the statement 48√ < 36√ is not true.
So, the correct statement among the given options is: 48√ > 36√.