Understanding the Square Root of 3249

The question asks for the square root of the number $3249$. The square root of a number is a value that, when multiplied by itself, gives the original number. Mathematically, finding the square root of $3249$ is finding a number $x$ such that $x^2 = 3249$. This is often denoted as $\sqrt{3249}$.

Analyzing the Given Options for Square Root

We are provided with four options for the square root of $3249$: $63$, $59$, $57$, and $67$. To find the correct square root, we can square each of these options and see which one results in $3249$.

Calculating and Finding the Square Root of 3249

Let's test each option by squaring it:

From the calculations above, we can see that squaring $57$ gives us $3249$. Therefore, the square root of $3249$ is $57$.

So, $\sqrt{3249} = 57$.

Square Root Calculation Revision

Additional Information on Square Roots

Finding the square root of a number like $3249$ can be done through various methods, including:

• Estimation: Since $50^2 = 2500$ and $60^2 = 3600$, $\sqrt{3249}$ must be between $50$ and $60$. The last digit of $3249$ is $9$, which means the square root must end in $3$ (since $3^2=9$) or $7$ (since $7^2=49$). This narrows down the possibilities between $50$ and $60$ to numbers ending in $3$ or $7$, like $53$ or $57$.
• Prime Factorization: Find the prime factors of the number. For every pair of identical prime factors, take one factor out. Multiply these factors to get the square root. For $3249 = 3 \times 1083 = 3 \times 3 \times 361 = 3 \times 3 \times 19 \times 19$. Taking one factor from each pair $(3 \times 3)$ and $(19 \times 19)$, we get $3 \times 19 = 57$.
• Long Division Method: A systematic algorithm similar to long division to find the square root digit by digit.

In this case, squaring the options provided the quickest path to verifying the correct square root of $3249$.