Question

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

Answer 1

To determine which statements are true, we can compare the square roots using rational approximations or simplified calculations.

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

   • \( \sqrt{36} = 6 \)
   • To approximate \( \sqrt{48} \): \[ \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \approx 4 \times 1.732 \approx 6.928 \] So, \( \sqrt{48} \approx 6.928 \) which is greater than \( \sqrt{36} = 6 \).

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

2. For \( \sqrt{49} \) and \( 7 \):

   • \( \sqrt{49} = 7 \) by definition. Hence, \( \sqrt{49} = 7 \).

3. Analyzing the statements:

   • \( \sqrt{48} > \sqrt{36} \) (True)
   • \( \sqrt{49} > 7 \) (False, because \( \sqrt{49} = 7 \))
   • \( \sqrt{48} < \sqrt{36} \) (False)
   • \( \sqrt{49} < 7 \) (False)

Based on these evaluations, the only true statement is:

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