Question

1. place the point on the number line that best approximates the location of \\(\sqrt{14}\\)
2. place the point on the number line that best approximates the location of \\(\sqrt{5}\\)
3. place the point on the number line that best approximates the location of \\(\sqrt{31}\\)

Explanation:

Question 11: Approximate $\boldsymbol{\sqrt{15}}$ on the number line

Step 1: Identify perfect squares around 15

We know $3^2 = 9$ and $4^2 = 16$. Since $9 < 15 < 16$, we have $3 < \sqrt{15} < 4$.

Step 2: Refine the approximation

15 is very close to 16 (since $16 - 15 = 1$), so $\sqrt{15}$ is very close to 4.

Question 12: Approximate $\boldsymbol{\sqrt{5}}$ on the number line

Step 1: Identify perfect squares around 5

We know $2^2 = 4$ and $3^2 = 9$. Since $4 < 5 < 9$, we have $2 < \sqrt{5} < 3$.

Step 2: Refine the approximation

5 is 1 unit away from 4 and 4 units away from 9, so $\sqrt{5}$ is closer to 2 (but between 2 and 3).

Question 13: Approximate $\boldsymbol{\sqrt{31}}$ on the number line

Step 1: Identify perfect squares around 31

We know $5^2 = 25$ and $6^2 = 36$. Since $25 < 31 < 36$, we have $5 < \sqrt{31} < 6$.

Step 2: Refine the approximation

31 is 6 units away from 25 and 5 units away from 36, so $\sqrt{31}$ is closer to 6 (but between 5 and 6).

Answer:

s:

1. Place the point very close to 4 (between 3 and 4, near 4).
2. Place the point between 2 and 3 (closer to 2).
3. Place the point between 5 and 6 (closer to 6).