Using a number line, what is the approximate value of 19−−√ ?(1 point) Responses 4.25 4.25 4.5 4.5 4.38 4.38 4.13

Answer 1

To approximate the value of \( \sqrt{19} \) using a number line, we can take note of the perfect squares around 19.

• The perfect square just below 19 is \( 16 \) (since \( 4^2 = 16 \)), which gives us \( \sqrt{16} = 4 \).
• The perfect square just above 19 is \( 25 \) (since \( 5^2 = 25 \)), which gives us \( \sqrt{25} = 5 \).

This tells us that \( \sqrt{19} \) is between 4 and 5.

To narrow it down further, we can look at the squares of numbers between 4 and 5:

• \( 4.4^2 = 19.36 \) (which is greater than 19)
• \( 4.3^2 = 18.49 \) (which is less than 19)

Thus, we know \( \sqrt{19} \) is between 4.3 and 4.4.

Next, we can check \( 4.35 \):

• \( 4.35^2 = 18.9225 \) (which is still less than 19)

And now \( 4.36 \):

• \( 4.36^2 = 19.0496 \) (which is greater than 19)

From this, we can conclude that \( \sqrt{19} \) is between 4.35 and 4.36, leading us to estimate that \( \sqrt{19} \approx 4.36 \).

Now looking at the options:

• 4.25
• 4.5
• 4.38
• 4.13

The closest approximate value from the options given for \( \sqrt{19} \) would be 4.38.

Answer 2

using a number line, what whole number and rational number to the nearest half is 105−−−√ between?(1 point) Responses between 9.5 and 10 between 9.5 and 10 between 11 and 11.5 between 11 and 11.5 between 10 and 10.5 between 10 and 10.5 between 10.5 and 11

Answer 3

To find the square root of \(105\) using a number line and to determine the intervals it falls into, we can check the perfect squares around \(105\).

1. The perfect square just below \(105\) is \(100\) (since \(10^2 = 100\)).
2. The perfect square just above \(105\) is \(121\) (since \(11^2 = 121\)).

This means that \( \sqrt{105} \) is between \( 10 \) and \( 11 \).

Next, we can refine our approximation by checking the squares of numbers between 10 and 11:

• For \(10.5\): \[ 10.5^2 = 110.25 \quad (\text{which is greater than } 105) \]

• For \(10.4\): \[ 10.4^2 = 108.16 \quad (\text{which is greater than } 105) \]

• For \(10.3\): \[ 10.3^2 = 106.09 \quad (\text{which is greater than } 105) \]

• For \(10.2\): \[ 10.2^2 = 104.04 \quad (\text{which is less than } 105) \]

Thus, we have determined that:

\[ 10.2^2 < 105 < 10.3^2 \]

This implies \( \sqrt{105} \) is between \( 10.2 \) and \( 10.3 \), and therefore, it is also between \( 10 \) and \( 10.5 \) when considering to the nearest half.

So, the best choice from the options provided would be:

Between 10 and 10.5.

Answer 4

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 48−−√<36−−√ start root 48 end root less than start root 36 end root 49−−√<7

Answer 5

Let's evaluate each statement one by one to determine which is true using rational approximations:

1. \( \sqrt{49} > 7 \)

   • \( \sqrt{49} = 7 \)
   • This statement is false since \( 7 \) is not greater than \( 7 \).

2. \( \sqrt{48} > \sqrt{36} \)

   • \( \sqrt{48} \) is approximately \( 6.93 \) (since \( 6.93^2 \approx 48 \)).
   • \( \sqrt{36} = 6 \)
   • This statement is true because \( 6.93 > 6 \).

3. \( \sqrt{48} < \sqrt{36} \)

   • As calculated above, \( \sqrt{48} \approx 6.93 \) and \( \sqrt{36} = 6 \).
   • This statement is false since \( 6.93 \) is not less than \( 6 \).

4. \( \sqrt{49} < 7 \)

   • As previously mentioned, \( \sqrt{49} = 7 \).
   • This statement is false since \( 7 \) is not less than \( 7 \).

Conclusion:

The only true statement among the options is:

\( \sqrt{48} > \sqrt{36} \).

Answer 6

can you simplify

Answer 7

Sure! Let’s break down the evaluation of each statement clearly and simply:

1. Statement: \( \sqrt{49} > 7 \)

   • Since \( \sqrt{49} = 7 \), this statement is false because \( 7 \) is not greater than \( 7 \).

2. Statement: \( \sqrt{48} > \sqrt{36} \)

   • \( \sqrt{48} \approx 6.93 \) (because \( 6.93^2 \approx 48 \)).
   • \( \sqrt{36} = 6 \).
   • This statement is true because \( 6.93 > 6 \).

3. Statement: \( \sqrt{48} < \sqrt{36} \)

   • We already know \( \sqrt{48} \approx 6.93 \) and \( \sqrt{36} = 6 \).
   • This statement is false because \( 6.93 \) is not less than \( 6 \).

4. Statement: \( \sqrt{49} < 7 \)

   • As established, \( \sqrt{49} = 7 \).
   • This statement is false because \( 7 \) is not less than \( 7 \).

True Statement:

The only true statement among the options is: \( \sqrt{48} > \sqrt{36} \).

Answer 8

Using rational approximations, what statement is true?(1 point) Responses 71−−√>61−−√ start root 71 end root greater than start root 61 end root 81−−√<9 start root 81 end root less than 9 71 −−−√<61−−√ start root 71 end root less than start root 61 end root 81−−√>9

Answer 9

Let's evaluate each statement one by one using rational approximations:

1. Statement: \( \sqrt{71} > \sqrt{61} \)

   • Approximate values:
     • \( \sqrt{71} \) is approximately \( 8.43 \) (since \( 8.43^2 \approx 71 \)).
     • \( \sqrt{61} \) is approximately \( 7.81 \) (since \( 7.81^2 \approx 61 \)).
   • Conclusion: True because \( 8.43 > 7.81 \).

2. Statement: \( \sqrt{81} < 9 \)

   • Since \( \sqrt{81} = 9 \), this statement is false because \( 9 \) is not less than \( 9 \).

3. Statement: \( \sqrt{71} < \sqrt{61} \)

   • As calculated above, \( \sqrt{71} \approx 8.43 \) and \( \sqrt{61} \approx 7.81 \).
   • Conclusion: False because \( 8.43 \) is not less than \( 7.81 \).

4. Statement: \( \sqrt{81} > 9 \)

   • Again, since \( \sqrt{81} = 9 \), this statement is false because \( 9 \) is not greater than \( 9 \).

True Statement:

The only true statement among the options is: \( \sqrt{71} > \sqrt{61} \).

Answer 10

Using rational approximations, what statement is true?(1 point) Responses 16 −−−√<4 start root 16 end root less than 4 12 −−−√<π start root 12 end root less than pi 12 −−−√ >π start root 12 end root greater than pi 16−−√>4

Answer 11

Let's evaluate each statement one by one to determine which one is true, using rational approximations:

1. Statement: \( \sqrt{16} < 4 \)

   • Since \( \sqrt{16} = 4 \), this statement is false because \( 4 \) is not less than \( 4 \).

2. Statement: \( \sqrt{12} < \pi \)

   • \( \sqrt{12} \) is approximately \( 3.46 \) (since \( 3.46^2 \approx 12 \)).
   • The value of \( \pi \) is approximately \( 3.14 \).
   • Therefore, \( 3.46 > 3.14 \), making this statement false.

3. Statement: \( \sqrt{12} > \pi \)

   • As calculated above, \( \sqrt{12} \approx 3.46 \) and \( \pi \approx 3.14 \).
   • Since \( 3.46 > 3.14 \), this statement is true.

4. Statement: \( \sqrt{16} > 4 \)

   • Since \( \sqrt{16} = 4 \), this statement is false because \( 4 \) is not greater than \( 4 \).

Conclusion:

The only true statement among the options is: \( \sqrt{12} > \pi \).