Estimate the sum of 5–√+7–√

Question

by performing increasingly small rational approximations. Approximate the sum to four decimal places. (1 point)
Responses

2.2361
2.2361

2.6458
2.6458

4.8819
4.8819

4.8818

Answer 1

To estimate the sum of $\sqrt{5}+\sqrt{7}$, we can find the square roots of 5 and 7 using increasingly smaller rational approximations.

1. Estimate $\sqrt{5}$:

   • We know that $2^2=4$ and $3^2=9$, so $\sqrt{5}$ is between 2 and 3.
   • To get more precise, we can try numbers between 2.2 and 2.3:
     • ${2.2}^2=4.84$
     • ${2.3}^2=5.29$

   Thus, $\sqrt{5}$ is between 2.2 and 2.3.

   Next, try $2.24$:

   • ${2.24}^2=5.0176$ (too high)
   • Next, try $2.23$:
   • ${2.23}^2=4.9729$ (too low)

   So $\sqrt{5}$ is between $2.23$ and $2.24$. We can take the average: $$\sqrt{5}\approx 2.236$$

2. Estimate $\sqrt{7}$:

   • We know that ${2.6}^2=6.76$ and ${2.7}^2=7.29$, so $\sqrt{7}$ is between 2.6 and 2.7.
   • Next, try $2.65$:
   • ${2.65}^2=7.0225$ (too high)
   • Next, try $2.64$:
   • ${2.64}^2=6.9696$ (too low)

   Thus, we find $\sqrt{7}$ is between $2.64$ and $2.65$. Using the average: $$\sqrt{7}\approx 2.645$$

3. Sum the Approximations: $$\sqrt{5}+\sqrt{7}\approx 2.236+2.645=4.881$$

Using a calculator or more precise approximations, we find that:

• $\sqrt{5}\approx 2.2361$
• $\sqrt{7}\approx 2.6458$

Therefore: $$\sqrt{5}+\sqrt{7}\approx 4.8819$$

Based on the options given, the best approximation to four decimal places is: 4.8819