Simplify the Square Root of 169 Easily and Quickly

The square root of 169 is a common mathematical problem that can be simplified easily. Here, we explore different methods to simplify √169.

Definition

The square root of a number is a value that, when multiplied by itself, gives the original number. Mathematically, the square root of 169 can be expressed as:

\[\sqrt{169}\]

We need to find a number \( q \) such that:

\[q \times q = 169\]

Calculation

To simplify √169, we can use the fact that 169 is a perfect square:

\[169 = 13 \times 13\]

Thus,

\[\sqrt{169} = 13\]

Methods of Simplification

1. Prime Factorization Method:

   169 can be expressed as the product of prime numbers:

   Taking the square root of both sides, we get:

   \[\sqrt{169} = \sqrt{13 \times 13} = 13\]

2. Repeated Subtraction Method:

   Subtract consecutive odd numbers from 169 until you reach 0:

   169 - 1 = 168

   168 - 3 = 165

   165 - 5 = 160

   ...

   Continue this process 13 times, indicating the square root is 13.

3. Long Division Method:

   Using long division, we can find the square root of 169 step-by-step:

   • Pair the digits of 169 from right to left: (1)(69).
   • Find the largest number whose square is less than or equal to 1. This is 1.
   • Double the quotient (1) and find the next digit (3) such that (2x10 + 3) x 3 is less than or equal to 69.
   • The quotient is 13.

Properties of the Square Root of 169

• Perfect Square: Since 169 is a perfect square, its square root is an integer.
• Rational Number: The square root of 169 is rational because it can be expressed as a fraction (13/1).
• Exponent Form: The square root of 169 can also be written as:

  \[169^{\frac{1}{2}} = 13\]

Examples

• Example 1: The area of a square is 169 square units. The length of the side is:

  Therefore, the side length is 13 units.

• Example 2: Simplify \(14 + \sqrt{169} - 7 \times 3\):

  \[14 + 13 - 21 = 6\]

Conclusion

In conclusion, the square root of 169 simplifies to 13 using various methods such as prime factorization, repeated subtraction, and long division.

Learn more about simplifying square roots and other mathematical concepts through practice and application of these methods.

The square root of 169 is a fundamental mathematical problem often encountered in various contexts, from academic exercises to practical applications. Understanding how to simplify square roots is essential for students and professionals alike. This guide will provide a comprehensive overview of different techniques used to simplify the square root of 169. These methods include prime factorization, repeated subtraction, and long division. By mastering these techniques, one can quickly and accurately determine that the square root of 169 is 13, enhancing their mathematical proficiency and confidence.

The square root of 169 is a fundamental mathematical concept, often encountered in various problems and applications. Simplifying the square root of 169 involves understanding that 169 is a perfect square. A perfect square is a number that can be expressed as the product of an integer with itself.

Mathematically, the square root of 169 is expressed as:

\(\sqrt{169} = 13\)

This is because \(13 \times 13 = 169\).

The process to determine the square root of 169 can be explained through several methods:

• Prime Factorization: Factorize 169 into its prime factors. Since \(169 = 13 \times 13\), it can be written as \(13^2\). Taking the square root of both sides gives \(\sqrt{169} = \sqrt{13^2} = 13\).
• Repeated Subtraction: Subtract consecutive odd numbers from 169 until you reach 0. This process takes 13 steps, confirming that \(\sqrt{169} = 13\).
• Long Division Method: Divide 169 using a systematic approach where the digits are grouped and divided step by step to find the square root. This method also results in \(\sqrt{169} = 13\).

The square root of 169 is also expressed in exponential form as \(169^{1/2}\), which equals 13. This property confirms that 169 is a rational number since its square root is an integer.

In conclusion, the square root of 169 is 13, a straightforward calculation due to 169 being a perfect square.

There are several methods to simplify the square root of 169, each offering a different approach to arrive at the same result. Below are the detailed steps for each method:

1. Prime Factorization Method

• Factorize 169 into prime numbers: \(169 = 13 \times 13\).
• Express 169 as a product of its prime factors: \(169 = 13^2\).
• Take the square root of both sides: \(\sqrt{169} = \sqrt{13^2} = 13\).

2. Repeated Subtraction Method

• Subtract consecutive odd numbers from 169 until reaching 0.
• 169 - 1 = 168
• 168 - 3 = 165
• 165 - 5 = 160
• 160 - 7 = 153
• 153 - 9 = 144
• 144 - 11 = 133
• 133 - 13 = 120
• 120 - 15 = 105
• 105 - 17 = 88
• 88 - 19 = 69
• 69 - 21 = 48
• 48 - 23 = 25
• 25 - 25 = 0
• This process takes 13 steps, hence the square root of 169 is 13.

3. Long Division Method

1. Write 169 as 1 and 69.
2. Find the largest number whose square is less than or equal to 1 (which is 1).
3. Place 1 as the first digit of the quotient.
4. Subtract the square of 1 from 1: 1 - 1 = 0. Bring down 69.
5. Double the quotient (1) and write it as 2.
6. Find a digit (3) to place after 2 to get a number whose product with 3 is less than or equal to 69 (23 * 3 = 69).
7. Subtract 69 from 69 to get 0. Place 3 as the next digit of the quotient.
8. The quotient is 13, hence the square root of 169 is 13.

4. Using a Calculator

• Enter 169 into the calculator.
• Press the square root (√) button.
• The display will show 13, indicating that the square root of 169 is 13.

5. Estimation and Refinement

• Estimate a number close to the square root of 169. Since 169 is between 144 (\(12^2\)) and 196 (\(14^2\)), the square root is between 12 and 14.
• Test the midpoint: \(13^2 = 169\).
• Since \(13^2\) exactly equals 169, the square root is 13.

Example 1: Area of a Square

Given a square with an area of 169 square units, we can determine the side length by taking the square root of 169. Since the square root of 169 is 13, each side of the square measures 13 units.

Example 2: Simplifying Mathematical Expressions

Simplify the expression \(14 + \sqrt{169} - 7 \times 3\).

• First, calculate the square root of 169: \( \sqrt{169} = 13 \).
• Then, substitute this value into the expression: \(14 + 13 - 7 \times 3\).
• Next, perform the multiplication: \(7 \times 3 = 21\).
• Finally, combine the terms: \(14 + 13 - 21 = 6\).

Therefore, the simplified result of the expression is 6.

Example 3: Solving for a Variable

Find the value of \( n \) if the square root of the sum of the number and 12 is 5.

• Start with the equation: \( \sqrt{n + 12} = 5 \).
• Square both sides to remove the square root: \( (\sqrt{n + 12})^2 = 5^2 \).
• This simplifies to: \( n + 12 = 25 \).
• Subtract 12 from both sides to solve for \( n \): \( n = 25 - 12 \).
• Thus, \( n = 13 \).

Example 4: Multiplication of Square Roots

Multiply \( \sqrt{25} \times \sqrt{16} \).

• Calculate each square root: \( \sqrt{25} = 5 \) and \( \sqrt{16} = 4 \).
• Multiply the results: \( 5 \times 4 = 20 \).

The product of \( \sqrt{25} \) and \( \sqrt{16} \) is 20.

Example 5: Rational Numbers

Determine whether the square root of 169 is a rational number.

• A rational number is defined as a number that can be expressed as the quotient or fraction of two integers.
• The square root of 169 is 13, which is a whole number.
• Since 13 can be expressed as \( \frac{13}{1} \), it is a rational number.

Therefore, the square root of 169 is a rational number.