What is the square root of 85?
Understand the Problem
The question is asking for the numerical value of the square root of 85. This involves calculating or approximating the square root of the number.
Answer
Approximately 9.22
Answer for screen readers
The square root of 85 can be approximated to around 9.22.
Steps to Solve
-
Identify the two perfect squares closest to 85
The perfect squares closest to 85 are 81 ($9^2$) and 100 ($10^2$). This indicates that the square root of 85 is between 9 and 10.
- Refine the approximation using division and averaging
A useful method for approximating square roots is to pick a number close to the expected root (let's say 9.5), and then use the formula:
$$ \text{New Approximation} = \frac{\text{Current Approximation} + \frac{85}{\text{Current Approximation}}}{2} $$
Start with 9.5:
$$ \text{New Approximation} = \frac{9.5 + \frac{85}{9.5}}{2} \approx \frac{9.5 + 8.947}{2} \approx \frac{18.447}{2} \approx 9.2235 $$
- Repeat the approximation step if necessary
Using 9.2235 as the new approximation:
$$ \text{New Approximation} = \frac{9.2235 + \frac{85}{9.2235}}{2} \approx \frac{9.2235 + 9.2167}{2} \approx \frac{18.4402}{2} \approx 9.2201 $$
So, the square root of 85 can be approximated to around 9.22.
The square root of 85 can be approximated to around 9.22.
More Information
The square root of a non-perfect square number like 85 is an irrational number, meaning it cannot be exactly expressed as a fraction, and its decimal representation goes on forever without repeating.
Tips
A common mistake is to stop refining the approximation too early, which can lead to a less accurate result. It's often beneficial to perform a few iterations of the division and averaging method to get a more precise approximation.
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