What is the square root of 156?
Understand the Problem
The question is asking for the numerical value of the square root of 156. To solve this, we will calculate the approximate value of the square root.
Answer
The approximate value of \( \sqrt{156} \) is \( 12.49 \).
Answer for screen readers
The approximate value of ( \sqrt{156} ) is ( \approx 12.49 ).
Steps to Solve
- Determine the nearest perfect squares
Identify the perfect squares closest to 156. The perfect squares are:
- $12^2 = 144$
- $13^2 = 169$
Since $144 < 156 < 169$, we know that $12 < \sqrt{156} < 13$.
- Estimate using a calculator or a numerical method
To find a more precise approximation of $\sqrt{156}$, we can use a calculator, or we can apply the method of successive averaging or estimation:
Let's average $12$ and $13$: $$ \text{Average} = \frac{12 + 13}{2} = 12.5 $$ Now we square $12.5$ to check: $$ 12.5^2 = 156.25 $$ Since $156.25$ is slightly more than $156$, we can try a value closer to $12.4$.
- Refine the estimate
Next, check $12.4$: $$ 12.4^2 = 153.76 $$ Check $12.5$ again: $$ 12.5^2 = 156.25 $$ Now, since $12.4 < \sqrt{156} < 12.5$, we can estimate the square root to be approximately $12.5$ but a bit lower.
- Use the average to narrow it down
We can try the average method again: $$ \text{Average} = \frac{12.4 + 12.5}{2} = 12.45 $$ Now we check $12.45^2$: $$ 12.45^2 = 155.6025 $$ Then, try $12.46$: $$ 12.46^2 = 155.8016 $$ Keep testing to narrow down closer values until you reach a satisfying precision.
- Final approximation
To finalize, the value can be conclusively approximated as: $$ \sqrt{156} \approx 12.49 $$
The approximate value of ( \sqrt{156} ) is ( \approx 12.49 ).
More Information
The square root of a number represents the value that, when multiplied by itself, gives that number. In real-world applications, precise calculations may be crucial, so using calculators is common.
Tips
- Rounding too early can lead to inaccurate results. Always keep extra decimal places until the final calculation.
- Confusing the order of roots and squares, it's essential to remember what you are calculating at each step.