What is the square root of 156?

Steps to Solve

1. 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 $12 < \sqrt{156} < 13$.

2. 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 $12.4$.

3. 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.

4. 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.

5. Final approximation

To finalize, the value can be conclusively approximated as: $$ \sqrt{156} \approx 12.49 $$