Using rational approximations, what is an inequality statement comparing √21 and √17? Enter <, >, or =.

Answer 1

To compare \(\sqrt{21}\) and \(\sqrt{17}\) using rational approximations, let's first find approximate values for both square roots.

1. \(\sqrt{21}\):

   • We know that \(4^2 = 16\) and \(5^2 = 25\), so \(\sqrt{21}\) is between 4 and 5.
   • A more precise approximation can be obtained by estimating: \[ 4.5^2 = 20.25 \quad \text{(too low)} \] \[ 4.6^2 = 21.16 \quad \text{(too high)} \]
   • Thus, \(\sqrt{21} \approx 4.58\).

2. \(\sqrt{17}\):

   • Similarly, since \(4^2 = 16\) and \(5^2 = 25\), \(\sqrt{17}\) is also between 4 and 5.
   • A more precise approximation: \[ 4.1^2 = 16.81 \quad \text{(too low)} \] \[ 4.2^2 = 17.64 \quad \text{(too high)} \]
   • So, \(\sqrt{17} \approx 4.12\).

Now, we compare the two approximations:

\[ \sqrt{21} \approx 4.58 \quad \text{and} \quad \sqrt{17} \approx 4.12 \]

Since \(4.58 > 4.12\), we conclude:

\[ \sqrt{21} > \sqrt{17} \]

Therefore, the inequality statement comparing \(\sqrt{21}\) and \(\sqrt{17}\) is:

\[ \sqrt{21} > \sqrt{17} \]

In the required format, the answer is:

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