Finding the Square Root of 6084

The question asks for the square root of the number 6084. Finding the square root of a number means finding a value that, when multiplied by itself, gives the original number. We are looking for a number $x$ such that $x^2 = 6084$. This value is denoted as $\sqrt{6084}$.

Methods to Calculate the Square Root of 6084

There are several methods to find the square root of a number like 6084, including:

• Prime Factorization Method: Breaking down the number into its prime factors.
• Long Division Method: A systematic way to calculate the square root.
• Estimation and Checking: Using known squares to estimate and then verifying.

For a multiple-choice question, the estimation and checking method using the provided options is often the quickest approach.

Using Estimation and Checking Options for $\sqrt{6084}$

We can check the square of each given option to see which one equals 6084.

• Option 1: 74
• Option 2: 82
• Option 3: 76
• Option 4: 78

Let's calculate the square of each option:

• For 74: $74 \times 74 = 5476$
• For 82: $82 \times 82 = 6724$
• For 76: $76 \times 76 = 5776$
• For 78: $78 \times 78 = ?$

Let's perform the multiplication for 78 multiplied by 78:

From the calculation, we see that $78 \times 78 = 6084$.

Therefore, the square of 78 is 6084.

This means that the square root of 6084 is 78.

So, $\sqrt{6084} = 78$.

Revision Table: Understanding Square Roots

Looking at this table, we see that 6084 is between 4900 ($70^2$) and 6400 ($80^2$). This confirms that the square root of 6084 should be between 70 and 80. The options provided (74, 82, 76, 78) fit this range, except for 82. Our calculation showed 78 is the correct value.

Additional Information: Perfect Squares and Unit Digits

A perfect square is an integer that is the square of an integer. For example, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are perfect squares ($1^2, 2^2, \ldots, 10^2$).

The unit digit of a perfect square can only be 0, 1, 4, 5, 6, or 9. The unit digit of 6084 is 4. This is consistent with it being a perfect square, as squares of numbers ending in 2 or 8 end in 4 ($2^2=4$, $8^2=64$). The options are 74 (ends in 4), 82 (ends in 2), 76 (ends in 6), and 78 (ends in 8). The squares of numbers ending in 4 or 6 end in 6. The squares of numbers ending in 2 or 8 end in 4. Since 6084 ends in 4, its square root must end in 2 or 8. Among the options, only 82 and 78 end in 2 or 8. We tested 78 and found it to be correct.