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: $${4.5}^2=20.25\text{(too low)}$$ $${4.6}^2=21.16\text{(too high)}$$
   • 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: $${4.1}^2=16.81\text{(too low)}$$ $${4.2}^2=17.64\text{(too high)}$$
   • So, $\sqrt{17}\approx 4.12$.

Now, we compare the two approximations:

$$\sqrt{21}\approx 4.58\text{and}\sqrt{17}\approx 4.12$$

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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