What is the Square Root of 1225?

The square root of 1225 is 35. This means that when 35 is multiplied by itself, the result is 1225.

Mathematical Representation

The square root of 1225 can be expressed in various forms:

• Radical Form: \( \sqrt{1225} \)
• Exponential Form: \( 1225^{1/2} \) or \( 1225^{0.5} \)

Methods to Find the Square Root of 1225

1. Prime Factorization

   To find the square root of 1225 using prime factorization:

   • Prime factorize 1225: \( 1225 = 5^2 \times 7^2 \)
   • Group the prime factors: \( \sqrt{1225} = \sqrt{5^2 \times 7^2} = 5 \times 7 = 35 \)

2. Long Division Method

   Steps to find the square root using the long division method:

   • Pair the digits from right to left: 12 | 25
   • Find a number whose square is less than or equal to 12: \( 3^2 = 9 \)
   • Bring down the next pair and continue the process to get 35 as the quotient.

3. Using a Calculator

   Simply enter 1225 into a calculator and use the square root function to get 35.

Applications and Examples

Understanding the square root of 1225 can be useful in various mathematical problems and real-life applications:

• Example 1

  Find the side length of a square with an area of 1225 square units.

  Solution: Side length = \( \sqrt{1225} = 35 \) units

• Example 2

  Calculate the cost of adding a protective layer around a square table with a surface area of 1225 square inches if the cost per inch is $0.50.

  Solution: Perimeter = \( 4 \times 35 = 140 \) inches, Cost = \( 140 \times 0.50 = 70 \) dollars

FAQs on Square Roots of 1225

• Is the square root of 1225 a rational number?

  Yes, the square root of 1225 is 35, which is a rational number.

• What is the square of 1225?

  The square of 1225 is \( 1225^2 = 1500625 \).

The square root of a number is a value that, when multiplied by itself, gives the original number. In this comprehensive guide, we will delve into the concept of the square root of 1225. Understanding square roots is fundamental in mathematics and has practical applications in various fields, including engineering, physics, and everyday problem-solving.

The square root of 1225 is a perfect example to illustrate these concepts. By definition, the square root of 1225 is the number which, when squared, results in 1225. Mathematically, this can be represented as:

\(\sqrt{1225} = 35\)

This is because \(35 \times 35 = 1225\). In this guide, we will explore different methods to calculate the square root of 1225, its properties, and various applications.

We will cover the following topics:

• Mathematical Representation
• Methods to Calculate the Square Root of 1225
• Properties of the Square Root of 1225
• Applications of the Square Root of 1225
• Examples Involving the Square Root of 1225
• Frequently Asked Questions

By the end of this guide, you will have a thorough understanding of the square root of 1225 and its significance in various contexts. Let's begin our journey into the world of square roots by understanding what exactly the square root of 1225 is and how it is determined.

The square root of a number is a value that, when multiplied by itself, gives the original number. In mathematical terms, the square root of x is denoted as √x. The square root of 1225 is a well-known value that can be calculated through various methods.

The square root of 1225 is 35. This is because 35 multiplied by itself equals 1225:

\( 35 \times 35 = 1225 \)

Mathematical Representation

• In radical form: \( \sqrt{1225} \)
• In exponential form: \( 1225^{\frac{1}{2}} \) or \( 1225^{0.5} \)

Steps to Calculate the Square Root of 1225

Here are some methods to determine the square root of 1225:

1. Prime Factorization Method:
   • Factorize 1225 into its prime factors: \( 1225 = 5^2 \times 7^2 \)
   • Take the square root of each prime factor: \( \sqrt{1225} = \sqrt{5^2 \times 7^2} \)
   • Simplify the expression: \( \sqrt{5^2} \times \sqrt{7^2} = 5 \times 7 = 35 \)
2. Long Division Method:
   • Pair the digits of 1225 from right to left: 12 | 25
   • Find the largest number whose square is less than or equal to 12, which is 3.
   • Subtract \( 3^2 = 9 \) from 12, bring down 25 to get 325.
   • Double the quotient (3) to get 6. Find a number (n) such that \( 6n \times n \leq 325 \). Here, \( n = 5 \).
   • Complete the calculation: \( 65 \times 5 = 325 \), thus the quotient is 35.
3. Using a Calculator:
   • Enter 1225 and press the square root function to get 35.
4. Estimation Method:
   • Identify two perfect squares between which 1225 lies (30² = 900 and 40² = 1600).
   • Since 1225 is closer to 900, estimate around 35, confirming that \( 35^2 = 1225 \).

Thus, through these methods, we confirm that the square root of 1225 is indeed 35.

The square root of 1225 can be calculated using various methods, each with its own steps and procedures. Here are the most common methods:

1. Prime Factorization

Prime factorization involves breaking down the number into its prime factors and then simplifying:

• Step 1: Find the prime factors of 1225.
• Step 2: 1225 = 5 × 5 × 7 × 7 = 5² × 7²
• Step 3: Group the factors into pairs: (5²) and (7²).
• Step 4: Take one factor from each pair: 5 and 7.
• Step 5: Multiply these factors: 5 × 7 = 35.

Thus, the square root of 1225 is 35.

2. Long Division Method

The long division method is a systematic way of finding the square root by dividing the number step-by-step:

• Step 1: Write 1225 and group the digits in pairs from the right: (12)(25).
• Step 2: Find a number that, when squared, is less than or equal to 12. This number is 3, as 3² = 9.
• Step 3: Subtract 9 from 12 to get the remainder 3, then bring down the next pair (25) to get 325.
• Step 4: Double the quotient (3) to get 6. Find a digit X such that (60+X)X is less than or equal to 325. This digit is 5, as 65 × 5 = 325.
• Step 5: Thus, the quotient is 35, so the square root of 1225 is 35.

3. Using a Calculator

Using a calculator is the quickest way to find the square root of 1225:

• Enter 1225 into the calculator.
• Press the square root (√) button.
• The calculator will display 35.

4. Estimation

Estimation involves finding two perfect squares between which 1225 lies and then estimating the square root:

• Step 1: Recognize that 1225 is between 1000 (32²) and 1600 (40²).
• Step 2: Since 1225 is closer to 1296 (36²) than to 1024 (32²), the square root is closer to 35.
• Step 3: Verify that 35² = 1225.

Thus, the estimated square root of 1225 is exactly 35.

Prime factorization is a method of expressing a number as the product of its prime factors. To find the square root of 1225 using prime factorization, follow these steps:

1. Find the prime factors of 1225: Start by dividing 1225 by the smallest prime number, which is 5.

\[1225 \div 5 = 245\]

2. Divide 245 by the smallest prime number, which is 5.

\[245 \div 5 = 49\]

3. Now, divide 49 by the smallest prime number, which is 7.

\[49 \div 7 = 7\]

4. Finally, divide 7 by itself since it is already a prime number.

\[7 \div 7 = 1\]

So, the prime factors of 1225 are:

\[1225 = 5 \times 5 \times 7 \times 7\]

5. Group the prime factors into pairs: Since we are looking for the square root, we group the factors into pairs.

\[1225 = (5 \times 5) \times (7 \times 7)\]

6. Take one number from each pair: To find the square root, take one number from each pair and multiply them.

\[\sqrt{1225} = 5 \times 7 = 35\]

Thus, the square root of 1225 is 35.

The long division method is a systematic way to find the square root of a number. Here's how you can find the square root of 1225 using this method:

1. Set up the number: Write 1225 as a pair of digits from right to left. For 1225, this will be \(12 \, | \, 25\).

2. Find the largest square less than or equal to the first pair (12): The largest perfect square less than 12 is 9 (3²), so write 3 as the first digit of the quotient.

   Subtract this square from the first pair:

   \(12 - 9 = 3\)

3. Bring down the next pair (25), making the new dividend 325.

4. Double the quotient: Double the current quotient (which is 3), making it 6. This number will be the beginning of your next divisor.

5. Find the next digit of the quotient: Find a digit to place next to 6 in the quotient (making it 6X) and as the last digit in the divisor (60X + X), such that \( (60 + X) \times X \leq 325 \). The digit X is 5 because \( 65 \times 5 = 325 \).

6. Complete the calculation: Subtract 325 from 325 to get a remainder of 0. Your quotient, which is now 35, is the square root of 1225.

Thus, using the long division method, we find that the square root of 1225 is 35.

Calculating the square root of 1225 using a calculator is straightforward and quick. Here are the steps to find the square root of 1225 using a calculator:

1. Turn on your calculator.

2. Enter the number 1225.

3. Press the square root (√) button. This button might be labeled as "sqrt" or just "√" depending on your calculator model.

4. Observe the result displayed on the screen. The calculator will show the square root of 1225, which is 35.

Using a calculator to find the square root of a number is one of the simplest methods available, ensuring quick and accurate results without the need for manual calculations.

Estimating the square root of a number is useful when you need a quick approximation without using a calculator. Here are the steps to estimate the square root of 1225:

1. Identify the nearest perfect squares: Determine which perfect squares 1225 is between. The perfect squares closest to 1225 are 900 (30²) and 1600 (40²).

2. Choose the closer perfect square: Since 1225 is closer to 900 than to 1600, we start with 30.

3. Refine the estimate: The next perfect square, 31², equals 961, and 32² equals 1024. Since 1225 is still larger than 1024, continue to the next perfect square.

   33² equals 1089, 34² equals 1156, 35² equals 1225.

4. Verify the estimate: Since 35² equals 1225, the square root of 1225 is exactly 35.

Thus, through estimation, we confirmed that the square root of 1225 is 35.

The square root of 1225 has several interesting properties, making it a valuable number in mathematics. Let's explore these properties in detail.

1. Rational or Irrational?

The square root of 1225 is a rational number. A rational number is a number that can be expressed as the quotient or fraction p/q of two integers, where p and q are integers and q is not zero. The square root of 1225 is 35, which can be written as 35/1, fitting the definition of a rational number.

2. Perfect Square

The square root of 1225 is a perfect square. A perfect square is an integer that is the square of another integer. In this case, 1225 is 35 squared (35 × 35). This means that 1225 is the product of 35 multiplied by itself, confirming that it is indeed a perfect square.

3. Positive and Negative Roots

Every positive real number has two square roots: one positive and one negative. Therefore, the square root of 1225 is both 35 and -35. However, the principal square root is considered to be the positive root, which is 35.

4. Radical and Exponential Forms

The square root of 1225 can be represented in different mathematical forms:

• Radical form: \( \sqrt{1225} \)
• Exponential form: \( 1225^{1/2} \)

5. Simplification and Prime Factorization

The prime factorization of 1225 is important for understanding its square root. The prime factors of 1225 are:

• 1225 = 5 × 5 × 7 × 7
• This can be written as 1225 = 5² × 7²
• Therefore, \( \sqrt{1225} = \sqrt{5^2 \times 7^2} = 5 \times 7 = 35 \)

6. Digit Properties

For any even number with n digits, the square root will have n/2 digits. For an odd number with n digits, the square root will have (n+1)/2 digits. Since 1225 has four digits, an even number, its square root (35) has two digits.

These properties help to understand the mathematical significance of the square root of 1225 and its applications in various calculations and problem-solving scenarios.

The square root of 1225 is a rational number. To understand why, we need to delve into the definition and properties of rational numbers.

A number is considered rational if it can be expressed as a fraction of two integers, where the numerator and the denominator are both integers and the denominator is not zero. In other words, a rational number can be written in the form p/q, where p and q are integers, and q ≠ 0.

The square root of 1225 can be determined by finding a number which, when multiplied by itself, gives the product of 1225. Mathematically, this is represented as:

√1225

To find this value, we can use the prime factorization method. The prime factorization of 1225 is:

1225 = 5 × 5 × 7 × 7

We can group the factors into pairs of equal factors:

1225 = (5 × 5) × (7 × 7)

Taking the square root of both sides, we get:

√1225 = √((5 × 5) × (7 × 7))

Which simplifies to:

√1225 = 5 × 7

Thus,

√1225 = 35

Since 35 is a whole number and can be expressed as 35/1, which is a fraction of two integers, it is clear that the square root of 1225 is a rational number. Therefore, we can conclude that the square root of 1225 is rational because it meets the criteria of being a number that can be expressed as a quotient of two integers.

A perfect square is a number that can be expressed as the square of an integer. In other words, if \( n \) is a perfect square, then there exists an integer \( m \) such that \( n = m^2 \).

In the case of 1225, it can be expressed as the square of 35:

\[
1225 = 35 \times 35 = 35^2
\]

This indicates that 1225 is indeed a perfect square. The integer 35, when multiplied by itself, results in 1225, fulfilling the definition of a perfect square.

To further verify, consider the prime factorization method:

• Prime factorize 1225: \( 1225 = 5^2 \times 7^2 \)
• Taking the square root of each factor: \( \sqrt{1225} = \sqrt{5^2 \times 7^2} = 5 \times 7 = 35 \)

Therefore, 1225 is confirmed to be a perfect square because its square root, 35, is an integer.

The square root of 1225, which is 35, finds applications in various fields. Below are some key areas where it is commonly used:

• Geometry and Construction:

  In construction, particularly in determining the lengths of diagonal supports or other structural elements, the square root is crucial. For instance, if a square plot has an area of 1225 square units, each side of the plot would measure 35 units (since \( \sqrt{1225} = 35 \)).

• Pythagorean Theorem:

  The square root function is essential in applications involving the Pythagorean theorem. For example, if the sum of the squares of two sides of a right triangle equals 1225, the hypotenuse would be 35 units long. This is calculated as:

  \[
  c = \sqrt{a^2 + b^2}
  \]
  where \( c \) is the hypotenuse.

• Distance Calculation:

  In coordinate geometry, the square root is used to determine the distance between two points. For instance, the distance formula in 2D space is given by:

  \[
  D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
  \]
  If the calculated value under the square root is 1225, the distance \( D \) would be 35 units.

• Physics and Engineering:

  Square roots are used in various physics and engineering equations. For example, to determine the period of a pendulum, which involves the square root of the length of the pendulum.

• Statistics:

  In statistics, the square root is often used in the calculation of standard deviation, which is a measure of the dispersion of a set of values. If the variance is 1225, the standard deviation would be:

  \[
  \sigma = \sqrt{1225} = 35
  \]

• Real-life Measurements:

  Square roots are also used in practical situations like determining dimensions of materials needed for construction, tailoring, and more.

Understanding the square root of 1225 can be made easier with practical examples. Here are a few scenarios where the square root of 1225 is applied:

• Example 1: Prime Factorization Method

  Paul wants to find the square root of 1225 using the prime factorization method. Follow these steps:

  1. Determine the prime factorization of 1225:

  \[ 1225 = 5 \times 5 \times 7 \times 7 = 5^2 \times 7^2 \]

  2. Group the prime factors in pairs:

  \[ 1225 = (5 \times 7)^2 \]

  3. Take one factor from each pair to find the square root:

  \[ \sqrt{1225} = \sqrt{(5 \times 7)^2} = 5 \times 7 = 35 \]

• Example 2: Area Calculation

  Ryan is working on a square-shaped table with a surface area of 1225 square inches. He wants to add a protective layer around the sides of the table.

  1. Find the length of one side of the table:

  \[ \text{Length of one side} = \sqrt{1225} = 35 \text{ inches} \]

  2. Calculate the total length of the protective layer required:

  \[ \text{Total length} = 4 \times 35 = 140 \text{ inches} \]

  3. Determine the cost of adding the protective layer if it costs $0.50 per inch:

  \[ \text{Total cost} = 140 \times 0.50 = $70 \]

• Example 3: Solving an Equation

  Solve the equation \( x^2 - 1225 = 0 \).

  1. Rewrite the equation:

  \[ x^2 = 1225 \]

  2. Find the square root of both sides:

  \[ x = \pm \sqrt{1225} = \pm 35 \]

  3. Thus, the solutions are:

  \[ x = 35 \text{ or } x = -35 \]

• Example 4: Nested Square Roots

  What is the value of \( \sqrt{\sqrt{1225}} \)?

  1. First, find the square root of 1225:

  \[ \sqrt{1225} = 35 \]

  2. Then, find the square root of 35:

  \[ \sqrt{35} \approx 5.92 \]

  3. Therefore:

  \[ \sqrt{\sqrt{1225}} \approx 5.92 \]

• 1. Is the Square Root of 1225 a Rational Number?

  Yes, the square root of 1225 is a rational number. A rational number is a number that can be expressed as the quotient or fraction of two integers. Since the square root of 1225 is 35, which is a whole number, it can be expressed as 35/1, making it a rational number.

• 2. What is the Square of 1225?

  The square of 1225 is calculated by multiplying 1225 by itself. Mathematically, it is represented as:

  \[ 1225 \times 1225 = 1500625 \]

  Thus, the square of 1225 is 1500625.

• 3. What are the Prime Factors of 1225?

  The prime factorization of 1225 involves breaking it down into its prime factors. The steps are as follows:

  1. 1225 is divisible by 5 (since it ends in 5). Divide 1225 by 5 to get 245.
  2. 245 is also divisible by 5. Divide 245 by 5 to get 49.
  3. 49 is divisible by 7 (since 7 x 7 = 49). Divide 49 by 7 to get 7.
  4. 7 is a prime number.

  Thus, the prime factorization of 1225 is:

  \[ 1225 = 5^2 \times 7^2 \]

The square root of a number is rational if it can be expressed as a fraction where both the numerator and the denominator are integers. In simpler terms, a number is rational if it can be written as a/b, where a and b are integers and b is not zero.

To determine if the square root of 1225 is rational, let's first find the square root:

\(\sqrt{1225} = 35\)

Since 35 is an integer, it can be written as a fraction (e.g., \(35/1\)), which means it is a rational number. Therefore, the square root of 1225 is rational.

Here are the steps to verify this:

• Find the square root of 1225: \(\sqrt{1225} = 35\)
• Check if 35 is an integer: Yes, it is.
• Write 35 as a fraction: \(35/1\)
• Since 35 can be expressed as a fraction of two integers, it is a rational number.

In conclusion, the square root of 1225 is indeed a rational number.

The square of a number is obtained by multiplying the number by itself. In mathematical terms, if n is a number, then the square of n is represented as n².

For the number 1225, we calculate its square as follows:

$$

1225
·
1225
=

1225
2

Using this calculation, we find that:

$$

1225
·
1225
=
1500625

Thus, the square of 1225 is 1,500,625. This means that when 1225 is multiplied by itself, it results in 1,500,625.

In summary, squaring a number essentially means multiplying the number by itself, and for 1225, the result of this operation is 1,500,625.

The prime factorization of a number involves breaking it down into its smallest prime factors. Let's find the prime factors of 1225 step by step:

1. Start with the smallest prime number, which is 2. Since 1225 is an odd number, it is not divisible by 2.

2. Move to the next prime number, which is 3. The sum of the digits of 1225 is 1+2+2+5 = 10, which is not divisible by 3, so 1225 is not divisible by 3.

3. Next, check 5. Since the last digit of 1225 is 5, it is divisible by 5:

   • 1225 ÷ 5 = 245

4. Now, we factor 245. Since the last digit of 245 is 5, it is also divisible by 5:

   • 245 ÷ 5 = 49

5. Next, we factor 49. Since 49 is not divisible by 2, 3, or 5, we try the next prime number, which is 7:

   • 49 ÷ 7 = 7
   • 7 ÷ 7 = 1

Therefore, the prime factors of 1225 are 5, 5, 7, and 7. This can be expressed as:

\[
1225 = 5^2 \times 7^2
\]

The square root of 1225 is a fundamental mathematical concept with various applications and interesting properties. As we have seen, the square root of 1225 is 35, which can be verified through different methods such as prime factorization, the long division method, and the use of calculators or estimation techniques.

Understanding that 1225 is a perfect square simplifies its square root calculation, confirming that it is a rational number. This is because 35, the square root, can be expressed as a simple fraction (35/1), fitting the definition of rational numbers.

The exploration of the square root of 1225 also illustrates the broader concept of square roots and their significance in mathematics. From solving quadratic equations to applications in geometry and various real-life scenarios, knowing how to determine and use square roots is essential.

In summary, the square root of 1225, which is 35, not only demonstrates the beauty of mathematics but also underscores the practical utility of mathematical principles in everyday life. By mastering these concepts, one can appreciate the elegance of numbers and their relationships, paving the way for further mathematical exploration and discovery.