To find the square root of 5329, we are looking for a number which, when multiplied by itself, equals 5329. The square root is the inverse operation of squaring a number.

Finding the Square Root of 5329

There are several methods to find the square root of a number like 5329. Let's explore a couple of common approaches.

Method 1: Estimation and Verification

We can estimate the range of the square root of 5329 by considering perfect squares of numbers close to the approximate value.

• We know that $70^2 = 70 \times 70 = 4900$.
• We know that $80^2 = 80 \times 80 = 6400$.

Since 5329 is between 4900 and 6400, its square root must be between 70 and 80. This narrows down the possibilities considerably.

Now, let's look at the last digit of 5329, which is 9. A perfect square ends in 9 if its square root ends in 3 (since $3^2=9$) or 7 (since $7^2=49$).

Considering the options provided, we are looking for a number between 70 and 80 that ends in 3 or 7. Only one option fits this description: 73.

Let's verify if 73 is indeed the square root of 5329 by multiplying 73 by itself:

$73 \times 73$

We can calculate this:

$73 \times 73 = (70 + 3) \times (70 + 3)$

$= 70 \times (70 + 3) + 3 \times (70 + 3)$

$= 70 \times 70 + 70 \times 3 + 3 \times 70 + 3 \times 3$

$= 4900 + 210 + 210 + 9$

$= 4900 + 420 + 9$

$= 5329$

The calculation confirms that $73^2 = 5329$. Therefore, the square root of 5329 is 73.

Method 2: Long Division Method for Square Roots

The long division method is a systematic way to find the exact square root of a number. Let's apply it to 5329.

Steps:

1. Group the digits in pairs from right to left. For 5329, this is (53)(29).
2. Find the largest number whose square is less than or equal to the leftmost group (53). $7^2 = 49$ and $8^2 = 64$. So, 7 is the number. Write 7 as the quotient and subtract $49$ from 53. The remainder is 4.
3. Bring down the next pair of digits (29) to the right of the remainder, making the new dividend 429.
4. Double the current quotient (7) and write it with a blank digit next to it. $2 \times 7 = 14$. We have 14_.
5. Find a digit to place in the blank ($14\_$) such that when the new number ($14\_$) is multiplied by the same digit, the product is less than or equal to 429.
   • If we try 1: $141 \times 1 = 141$ (too small)
   • If we try 2: $142 \times 2 = 284$ (too small)
   • If we try 3: $143 \times 3 = 429$ (exact match!)
6. Write 3 as the next digit in the quotient. Subtract $429$ from $429$. The remainder is 0.

Since the remainder is 0, the square root is the quotient obtained, which is 73.

Both methods confirm that the square root of 5329 is 73.

Summary of Square Root Finding

Finding the square root of 5329 involves identifying the number that squares to 5329. Using estimation based on nearby squares and considering the last digit, combined with verifying the result ($73 \times 73 = 5329$), or using the systematic long division method, leads to the answer 73.

Revision Table: Key Concepts

Additional Information on Square Roots

Square roots are fundamental in mathematics and have applications in various fields.

• Principal Square Root: For a positive number, there are two square roots (a positive and a negative one). For example, both 73 and -73 squared equal 5329. However, the symbol $\sqrt{}$ usually denotes the principal (non-negative) square root.
• Applications: Square roots are used in geometry (e.g., Pythagorean theorem to find the length of a side of a right triangle), physics, statistics, and many other areas.
• Irrational Square Roots: Not all numbers have integer square roots. For example, the square root of 2 ($\sqrt{2}$) is an irrational number, meaning it cannot be expressed as a simple fraction.

Understanding how to calculate square roots, whether of perfect squares like 5329 or using approximations for non-perfect squares, is an essential mathematical skill.