Estimate the square root between two consecutive whole numbers.
$\sqrt{138}$

• A. Between $11$ and $12$
• B. Between $10$ and $11$
• C. Between $10$ and $12$
• D. Between $11$ and $13$

Answer: (A)

Verified step by step guidance

1. Identify the number under the square root, which is 138 in this case, and understand that we want to find two consecutive whole numbers between which \( \sqrt{138} \) lies.
2. Recall that the square root of a number \( n \) is a value \( x \) such that \( x^2 = n \). So, we need to find two whole numbers \( a \) and \( a+1 \) where \( a^2 < 138 < (a+1)^2 \).
3. Find the perfect squares closest to 138 by checking squares of whole numbers near it. For example, calculate \( 11^2 \) and \( 12^2 \) to see if 138 lies between these two squares.
4. Compare the values: if \( 11^2 < 138 < 12^2 \), then \( \sqrt{138} \) lies between 11 and 12. If not, check the next pair of consecutive squares until you find the correct interval.
5. Once you find the two consecutive whole numbers \( a \) and \( a+1 \) such that \( a^2 < 138 < (a+1)^2 \), you have successfully estimated the square root of 138 to be between these two numbers.