Estimating a Square Root to the Hundrediths Pla Estimate \( \sqrt{50} \) to the hundredths place. 1. Estimate between two whole numbers: \[ 7^{2}=49,8^{2}=64 \] 2. Estimate further to the tenths place: \( 7.0^{2}=49.0,7.1^{2}=50.41 \) 3. Estimate further to the hundredths place: The \( \sqrt{50} \) is between \( \square \) \( \sqrt{ } 7.07 \times \) and \( \square \) I. \[ \begin{array}{l} 7.06 \\ \times 7.07 \\ 7.08 \\ 7.09 \end{array} \]

To estimate \( \sqrt{50} \) further to the hundredths place, you evaluate the squares of numbers like \( 7.07 \), \( 7.08 \), \( 7.09 \), and perhaps even numbers in between. Through testing, you find: - \( 7.07^2 \approx 49.5049 \) - \( 7.08^2 \approx 49.9664 \) - \( 7.09^2 \approx 50.0641 \) From these values, you can deduce that \( \sqrt{50} \) is between \( 7.07 \) and \( 7.09 \). If we continue this pattern, we find that \( 7.08 \) squares very close to 50, suggesting that the closest estimation would lead us to conclude that \( \sqrt{50} \approx 7.07 \) to the hundredths place is reasonable. Fun fact: The method of refining estimates can be linked to ancient civilizations, like the Babylonians, who used similar techniques over 3,000 years ago to derive square roots! While it might seem a bit puzzling today, their approach laid the foundation for mathematical estimation techniques we still use. When estimating square roots, a common mistake is to overlook checking the squares of surrounding numbers closely enough. Don't just stop at the tenths; check a few values in between to ensure your approximation is not just close, but spot-on! Always error on the side of checking a bit more; it can save you from rounding pitfalls.