Using a number line, what is the approximate value of 19−−√?
We can start by noticing that $16=4^2<19<5^2=25$. Therefore, $\sqrt{19}$ is between 4 and 5. Dividing the number line from 4 to 5 into 10 equal segments gives a distance of 0.1 on each segment (since 4.1, 4.2, 4.3, etc. must each be 0.1 units away from 4). We see that 19 is closer to 4 than to 5, so 19−−√ is closer to 4 than to 5, and hence it is closer to $\boxed{4}$.
To find the approximate value of 19−−√, we can use a number line.
Step 1: Start by placing 19 on the number line.
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10 15 19 20 25 30 35 40
Step 2: We know that the square root of a number is the value that, when multiplied by itself, gives the original number. So, we are looking for the value that, when squared, gives us 19.
Step 3: Start with a guess. Let's try 4.
19 = 4 × 4 = 16 (too small)
Step 4: Since 4 is too small, let's try a larger number. Let's try 5.
19 = 5 × 5 = 25 (too big)
Step 5: Now, let's try a number between 4 and 5. Let's try 4.5.
19 = 4.5 × 4.5 = 20.25 (too big)
Step 6: The square root of 19 is between 4 and 4.5. Let's try a number between 4 and 4.5. Let's try 4.2.
19 ≈ 4.2 × 4.2 ≈ 17.64 (a little too small)
Step 7: Since 4.2 is a little too small, let's try a larger number. Let's try 4.3.
19 ≈ 4.3 × 4.3 ≈ 18.49 (a little too small)
Step 8: Since 4.3 is a little too small, let's try an even larger number. Let's try 4.4.
19 ≈ 4.4 × 4.4 ≈ 19.36 (a little too big)
Step 9: The square root of 19 is between 4.3 and 4.4, so our approximate value will be the average of these two numbers.
4.3 + 4.4 = 8.7
8.7 ÷ 2 ≈ 4.35
Therefore, the approximate value of 19−−√ is approximately 4.35.