Square Root Calculation Explained

To find the square root of a given number, 28153636, we are looking for a number that, when multiplied by itself, results in 28153636. A square root can be positive or negative, as both a positive number squared and a negative number squared yield a positive result (e.g., $2^2 = 4$ and $(-2)^2 = 4$).

Square Root Estimation

• We can first estimate the range in which the square root of 28153636 will fall.
• Consider the nearest perfect squares for numbers in the tens of millions:
• The square of 5000 is $5000^2 = 25,000,000$.
• The square of 6000 is $6000^2 = 36,000,000$.
• Since 28153636 is greater than 25,000,000 but less than 36,000,000, its square root must be a number between 5000 and 6000. All the given options ($\pm 5806, \pm 5906, \pm 5306, \pm 5406$) fit within this estimated range.

Units Digit Analysis for Square Root

• The given number, 28153636, ends with the digit 6.
• When a number is squared, its last digit is determined by the last digit of the original number. For a perfect square to end with 6, its square root must end with either 4 (because $4 \times 4 = 16$) or 6 (because $6 \times 6 = 36$).
• Observing the provided options, all of them end with the digit 6. This means that using the units digit alone does not help us eliminate any of the choices directly.

Verifying the Correct Square Root Option

• Given the multiple-choice options, the most straightforward approach is to test each option by squaring it to see which one results in 28153636.
• Let's test the option $\pm 5306$. We need to calculate $5306^2$.
• We can expand this using the algebraic identity $(a+b)^2 = a^2 + 2ab + b^2$. Let $a = 5300$ and $b = 6$.
• First, calculate $a^2$:
• $a^2 = 5300^2 = 53^2 \times 100^2 = 2809 \times 10000 = 28,090,000$
• Next, calculate $2ab$:
• $2ab = 2 \times 5300 \times 6 = 10600 \times 6 = 63,600$
• Finally, calculate $b^2$:
• $b^2 = 6^2 = 36$
• Now, add these three components together to find the total square:
• $5306^2 = a^2 + 2ab + b^2 = 28,090,000 + 63,600 + 36 = 28,153,636$
• Since $5306^2 = 28153636$, this confirms that $\pm 5306$ is indeed the correct square root of 28153636.

Final Square Root Answer

Therefore, the square root of 28153636 is $\pm 5306$.