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Square Root of 13: Step-by-Step Guide for Students

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How to Calculate and Simplify the Square Root of 13

The Square Root of 13 is a commonly encountered concept in mathematics, especially for students preparing for board exams and competitive tests like JEE Main, NEET, or Olympiads. Knowing how to find and use √13 is important in number theory, algebra, and geometry, and helps develop strong calculation and problem-solving skills.


What is the Square Root of 13?

The square root of 13, written as √13, is the value that, when multiplied by itself, gives the number 13. In mathematical notation, if \( x = \sqrt{13} \), then \( x^2 = 13 \). Since 13 is not a perfect square, its square root does not result in a whole number.

The decimal value of √13 is approximately 3.605551, and it continues infinitely without terminating or repeating a pattern.


Key Properties of the Square Root of 13

  • Symbolic (Radical) Form: √13
  • Decimal Form: 3.605551...
  • Simplest Form: Cannot be simplified further; stays as √13
  • Rational/Irrational: Irrational number
  • Between which integers: Lies between 3 and 4 (since \(3^2 = 9\), \(4^2 = 16\))

Representation and Decimal Expansion

√13 is an irrational number, meaning its decimal expansion is non-terminating and non-repeating. For practical problems, it's usually rounded to 3.605, 3.61, or 3.606, depending on the required accuracy.

Radical FormDecimal ApproximationBetween Which Integers
√13 3.605551... 3 (9) < √13 < 4 (16)

Is the Square Root of 13 Rational or Irrational?

The square root of 13 is irrational. Here’s why:

  • It cannot be written as a fraction \( \frac{p}{q} \) where both p and q are integers and q ≠ 0.
  • The decimal expansion of √13 neither terminates nor repeats any pattern.
  • Prime factorization of 13 gives only one 13 (a single prime, with no pairs), so its root can’t be expressed exactly as a rational number.

How to Find the Square Root of 13

There are several ways to estimate or calculate √13:

  • Prime Factorization: Not effective here, as 13 is prime and not a perfect square.
  • Estimation: Find perfect squares near 13 (9 and 16), estimate between their roots (3 and 4).
  • Long Division Method: Most reliable for non-perfect squares. Here’s how you find √13 step-by-step:
  1. Start with 13. Put it as 13.000000 to get decimals.
  2. Group digits in pairs from the decimal. For 13: 13.00 00 00
  3. Find the largest integer whose square is ≤ 13 (3 × 3 = 9).
  4. Subtract 9 from 13: remainder 4.
  5. Bring down the next pair of zeros: new dividend = 400.
  6. Double the quotient (3 × 2 = 6), and consider the next digit (let’s say x): 6x × x ≤ 400
  7. The largest x is 6, because 66 × 6 = 396 ≤ 400.
  8. Continue the process for greater decimal accuracy.

The quotient emerging from this method gives you √13 to the required number of decimal places. The process shows that √13 ≈ 3.605551.


Worked Example: Calculating √13 by Long Division

  1. Group and estimate: √13 → 3 is the closest integer (because 3² = 9, 4² = 16).
  2. Subtract: 13 – 9 = 4, bring down 00 → 400.
  3. Double 3 → 6_. Now, 66 × 6 = 396 (best fit under 400). Write 6 after decimal.
  4. Subtract: 400 – 396 = 4, bring down 00 → 400.
  5. Double 36 (ignore decimal) → 72_0. 720 × 0 = 0 under 400, so next digit is 0.
  6. Continue: bring down next pair of 0s, form new divisor by doubling previous digits, repeat to get more decimal places.

Thus, in steps, √13 = 3.605551...


Simplification & Radical Form

  • √13 is already in its simplest radical form. As 13 is a prime number, you cannot simplify it further (unlike √12 = 2√3).
  • Exponential form: \( 13^{1/2} \) or 130.5

Application Examples

Let’s see where √13 appears in real maths problems:

  1. Geometry: If a square field has an area of 13 m², then each side is √13 m (~3.61 m).
  2. Pythagoras Theorem: In a right triangle, if one side is 2 units and the hypotenuse is 5 units, use the formula \( a^2 + b^2 = c^2 \).\br So, \( a^2 + 2^2 = 5^2 \implies a^2 = 25 - 4 = 21 \) (not 13, but if a² = 13, then a = √13)
  3. Algebra: Solve \( x^2 - 13 = 0 \); solutions: \( x = ±\sqrt{13} \).

Practice Problems

  • Estimate the value of √13 to two decimal places.
  • Is the square root of 13 rational or irrational? Explain why.
  • Simplify: √13 × 2
  • If the side of a square is √13 cm, what is its area?
  • Find two numbers between which √13 lies.
  • Solve: \( x^2 = 13 \)

Common Mistakes to Avoid

  • Assuming √13 is rational or can be simplified to a simpler radical expression.
  • Believing √13 is exactly 3.6 or 3.61; Always clarify that the decimal keeps going and doesn’t repeat or terminate.
  • Mixing up the concept of perfect squares and non-perfect squares.
  • Incorrectly using prime factorization for non-perfect squares.

Real-World Applications

You may use √13 in finding diagonal lengths, distances (like in the distance formula in geometry), or to solve quadratic equations in physics and engineering. For instance, if you measure the diagonal of a square with area 13 units², the diagonal will be √(2×13) = √26 units. At Vedantu, you’ll find many more square root applications in geometry, algebra, and exam problems.


In summary, the Square Root of 13 is an irrational number, approximately 3.605551, and cannot be simplified further. Understanding how to estimate, calculate, and apply √13 is crucial for lesson problems and competitive exams. At Vedantu, we help you master concepts like these with stepwise examples and practice questions, so you can solve questions confidently in any exam!


For similar concepts, you can explore: Square Root of 12, Square Root of 14, Square Root Finder, and Irrational Numbers.


FAQs on Square Root of 13: Step-by-Step Guide for Students

1. What is the square root of 13?

The square root of 13, denoted as √13, is an irrational number approximately equal to 3.605551. It lies between the integers 3 and 4 because 3² = 9 and 4² = 16. This means it cannot be expressed as a simple fraction and its decimal representation continues infinitely without repeating.

2. How do you calculate the square root of 13?

The square root of 13 can be calculated using several methods: approximation, using a calculator, or the long division method. The long division method involves a step-by-step process of grouping digits and estimating digits to achieve the desired decimal accuracy. Calculators provide a quick approximation.

3. Is the square root of 13 rational or irrational?

√13 is an irrational number. This is because it cannot be expressed as a fraction of two integers (a/b where 'a' and 'b' are integers and b≠0). Its decimal representation is non-terminating and non-repeating.

4. How to find the value of √13?

To find the value, you can use a calculator or the long division method. Calculators directly give an approximate decimal value (3.605551). The long division method provides a step-by-step approach to finding the decimal approximation to any desired level of accuracy.

5. What is the value of √13 between?

The value of √13 is between 3 and 4. This is because 3 squared (3²) equals 9, and 4 squared (4²) equals 16, and 13 lies between 9 and 16.

6. How to find the square of 13?

The square of 13 (13²) is simply 13 multiplied by itself: 13 x 13 = 169. This is the reverse operation of finding the square root.

7. Can the square root of 13 be simplified?

No, the square root of 13 (√13) cannot be simplified further because 13 is a prime number. It has no perfect square factors other than 1.

8. What is the square root of 13 in radical form?

The square root of 13 in radical form is simply written as √13. This is its simplest form because 13 is a prime number and has no perfect square factors.

9. What is the square root of 13 simplified?

√13 is already in its simplest form. Because 13 is a prime number, it cannot be simplified further using prime factorization or other methods of simplification for square roots.

10. How does the decimal expansion of √13 behave infinitely?

The decimal expansion of √13 is non-terminating and non-repeating. This is a characteristic of irrational numbers. It continues infinitely without ever settling into a repeating pattern of digits.

11. Is √13 a rational number?

No, √13 is not a rational number; it is an irrational number. Rational numbers can be expressed as a fraction (a/b) where a and b are integers, and b is not zero. √13 cannot be expressed in this form.

12. Square root of 13 by division method?

The long division method for finding the square root of 13 involves a step-by-step process of estimating digits and refining the approximation. It's a more manual way to calculate the approximate value compared to using a calculator. The process involves pairing digits and progressively improving the accuracy of the square root approximation.