To estimate the square root of 150 to the nearest integer, we can first identify the perfect squares around 150.
The perfect squares close to 150 are:
- \(12^2 = 144\)
- \(13^2 = 169\)
Since \(144 < 150 < 169\), it follows that:
\[ 12 < \sqrt{150} < 13 \]
Next, we can refine our estimate. Let's calculate the square of 12.5:
\[ 12.5^2 = 156.25 \]
Since \(156.25 > 150\), we know \(\sqrt{150} < 12.5\).
Now, let's try 12.3 and 12.4:
- For \(12.3\):
\[ 12.3^2 = 151.29 \]
- For \(12.4\):
\[ 12.4^2 = 153.76 \]
Both calculations show that \(12.3^2 > 150\) and \(12.4^2 > 150\).
Let's try a value lower than 12.3:
- For \(12.2\):
\[ 12.2^2 = 148.84 \]
Now we can summarize that:
\[ 12.2^2 < 150 < 12.3^2 \]
This indicates that \(12.2 < \sqrt{150} < 12.3\).
To conclude, since the estimates show that \(\sqrt{150}\) is closer to \(12.2\), and since it falls between 12.2 and 12.3, we round to the nearest integer, which is 12.
Thus, the estimated square root of 150 to the nearest integer is:
\[ \boxed{12} \]
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