Finding the Square Root of 27225

The question asks us to find the square root of the number 27225. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because $3 \times 3 = 9$. We are looking for a number $x$ such that $x^2 = 27225$.

Methods to Calculate Square Roots

There are several ways to find the square root of a number, such as the long division method, prime factorization, or estimation, especially when multiple-choice options are provided.

Calculating $\sqrt{27225}$ Using Estimation and Verification

Since the number 27225 ends with the digit 5, its square root must also end with the digit 5. This is because the square of any number ending in 5 will always end in 25.

Let's look at the given options, all of which end in 5:

1. 165
2. 175
3. 155
4. 145

We can estimate the range of the square root. Consider the numbers formed by the digits before the last two (272). We need to find the squares of numbers ending in 5 that are close to $\sqrt{27225}$.

• $100^2 = 10000$
• $200^2 = 40000$

Since 27225 is between 10000 and 40000, its square root is between 100 and 200.

Let's check the squares of numbers ending in 5 around the range suggested by the options:

• Consider 145: $145^2 = (140 + 5)^2 = 140^2 + 2 \times 140 \times 5 + 5^2 = 19600 + 1400 + 25 = 21025$. This is less than 27225.
• Consider 155: $155^2 = (150 + 5)^2 = 150^2 + 2 \times 150 \times 5 + 5^2 = 22500 + 1500 + 25 = 24025$. This is also less than 27225.
• Consider 165: $165^2 = (160 + 5)^2 = 160^2 + 2 \times 160 \times 5 + 5^2 = 25600 + 1600 + 25 = 27225$. This matches the number.

We have found that $165 \times 165 = 27225$. Therefore, the square root of 27225 is 165.

For completeness, let's also check the next option:

• Consider 175: $175^2 = (170 + 5)^2 = 170^2 + 2 \times 170 \times 5 + 5^2 = 28900 + 1700 + 25 = 30625$. This is greater than 27225.

Verification of the Square Root

To verify our answer, we simply square the number 165:

$\qquad 165 \times 165 = 27225$

This confirms that the square root of 27225 is indeed 165.

Revision Table: Square Roots and Squares

Additional Information: Perfect Squares

A perfect square is an integer that is the square of an integer. In other words, it is the product of an integer with itself. For example, 9 is a perfect square because $9 = 3 \times 3 = 3^2$. The number 27225 is a perfect square because its square root, 165, is an integer.

Properties of perfect squares:

• A perfect square can only end with the digits 0, 1, 4, 5, 6, or 9.
• If a perfect square ends in 0, it must end in 00.
• If a perfect square ends in 5, it must end in 25.
• If a perfect square ends in 1, 4, 9, or 6, the digit before the last digit must be even, except when the number ends in 6, where the digit before the last digit must be odd (e.g., 16, 36, 256).
• The sum of the digits of a perfect square (repeatedly added until a single digit is obtained) can only be 1, 4, 7, or 9. For 27225: $2+7+2+2+5 = 18$, $1+8=9$. This aligns with the property.