Question

approximate \\(\sqrt{88}\\) by following the steps below.\\(\sqrt{88}\\) must lie between the whole numbers 9 and 10 because \\(9^2 = 81\\) and \\(10^2 = 100\\), and 88 lies between these values. drag \\(\sqrt{88}\\) based on your estimate above: (image of a number line from 0 to 10 with \\(\sqrt{88}\\) near 9) to one decimal place, \\(\sqrt{88}\\) must lie between 9.3 and 9.4 because \\(9.3^2 = 86.49\\) and \\(9.4^2 = 88.36\\), and 88 lies between these values. drag \\(\sqrt{88}\\) based on your estimate above: (image of a number line from 9 to 10 with \\(\sqrt{88}\\) near 9.3) to two decimal places, \\(\sqrt{88}\\) must lie between \\(\square\\) and \\(\square\\)

Explanation:

Step1: Find the lower bound

We know that \(9.3^2 = 86.49\) and \(9.4^2 = 88.36\). Now we start checking values between 9.3 and 9.4. Let's first check \(9.38^2\):
\(9.38^2=(9 + 0.38)^2=9^2+2\times9\times0.38 + 0.38^2=81+6.84 + 0.1444 = 87.9844\)

Step2: Find the upper bound

Now check \(9.39^2=(9 + 0.39)^2=9^2+2\times9\times0.39+0.39^2=81 + 7.02+0.1521 = 88.1721\)
Since \(87.9844<88<88.1721\), so \(\sqrt{88}\) lies between \(9.38\) and \(9.39\) when rounded to two decimal places.

Answer:

\(9.38\) and \(9.39\)