Question

Using rational approximations, what statement is true?(1 point) Responses 48−−√<36−−√ start root 48 end root less than start root 36 end root 48−−√>36−−√ start root 48 end root greater than start root 36 end root 49−−√<7 start root 49 end root less than 7 49−−√>7

Answer 1

To determine the truth of the provided statements using rational approximations of the square roots, we can first calculate the approximate values of the square roots involved.

1. \( \sqrt{48} \) and \( \sqrt{36} \):

   • \( \sqrt{36} = 6 \)
   • \( \sqrt{48} \approx 6.93 \) (since \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \) and \( \sqrt{3} \approx 1.732 \), thus \( 4 \times 1.732 \approx 6.928 \))

   In this case, \( \sqrt{48} \approx 6.93 > 6 = \sqrt{36} \), which shows that \( \sqrt{48} > \sqrt{36} \).

2. \( \sqrt{49} \):

   • \( \sqrt{49} = 7 \)

   • From this, we have \( \sqrt{49} = 7 \).

Now let's analyze the statements:

• \( \sqrt{48} < \sqrt{36} \): False
• \( \sqrt{48} > \sqrt{36} \): True
• \( \sqrt{49} < 7 \): False
• \( \sqrt{49} > 7 \): False
• \( \sqrt{49} < 7 < \sqrt{49} \): Also, not applicable.

Thus, considering the values, the true statement is:

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