Square Root of 5/3
2026-02-28 06:21 Diff
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Last updated on August 5, 2025

If a number is multiplied by itself, the result is a square. The inverse of the square is a square root. The square root is used in various fields including engineering, finance, etc. Here, we will discuss the square root of 5/3.

What is the Square Root of 5/3?

The square root is the inverse of the square of a number. 5/3 is not a perfect square. The square root of 5/3 is expressed in both radical and exponential form. In radical form, it is expressed as √(5/3), whereas in exponential form it is expressed as (5/3)^(1/2). √(5/3) ≈ 1.29099, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 5/3

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect squares. Instead, methods such as the long-division method and approximation method are used. Let us now learn the following methods:

  • Long division method
  • Approximation method

Square Root of 5/3 by Long Division Method

The long division method is particularly used for non-perfect squares. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: Convert 5/3 to a decimal: 5/3 ≈ 1.6667.

Step 2: Group the digits of 1.6667 in pairs from the decimal point outward, adding zeros if necessary: 1.66 | 67.

Step 3: Find the largest integer n whose square is less than or equal to 1. The integer is 1 (since 1^2 = 1).

Step 4: Subtract 1 from 1 to get the remainder 0. Bring down the next pair of digits 66 to get 066.

Step 5: Double the quotient obtained and use it as the new divisor's base. The divisor becomes 20.

Step 6: Determine the value of the next digit of the quotient (n) such that 20n × n ≤ 66. Here, n = 3, since 203 × 3 = 609.

Step 7: Subtract 609 from 660 to get the remainder 51, and bring down the next pair of zeros to get 5100.

Step 8: Continue these steps to get the decimal expansion of √(5/3).

So, √(5/3) ≈ 1.29099.

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Square Root of 5/3 by Approximation Method

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 5/3 using the approximation method.

Step 1: Estimate a rough range. Since 5/3 ≈ 1.67, check between √1 and √2.

Step 2: Refine the range using closer approximations. √1.5 ≈ 1.2247 and √2 ≈ 1.4142.

Step 3: Use linear interpolation or trial and error between 1.2247 and 1.4142 to narrow down closer to √1.67.

Step 4: Using interpolation, √(5/3) ≈ 1.29.

Common Mistakes and How to Avoid Them in the Square Root of 5/3

Students often make mistakes while finding square roots, such as forgetting about the negative square root or skipping steps in the long division method. Let's look at some common mistakes in detail.

Problem 1

Can you help Max find the area of a square box if its side length is given as √(5/3)?

Okay, lets begin

The area of the square is approximately 1.667 square units.

Explanation

The area of the square = side^2.

The side length is given as √(5/3).

Area of the square = side^2 = √(5/3) × √(5/3) = 5/3 ≈ 1.667.

Therefore, the area of the square box is approximately 1.667 square units.

Well explained 👍

Problem 2

A square-shaped garden measuring 5/3 square meters is created; if each side is √(5/3), what will be the area of half of the garden?

Okay, lets begin

Approximately 0.8335 square meters.

Explanation

We can divide the given area by 2 as the garden is square-shaped.

Dividing 5/3 by 2 gives us approximately 0.8335.

So, half of the garden measures approximately 0.8335 square meters.

Well explained 👍

Problem 3

Calculate √(5/3) × 4.

Okay, lets begin

Approximately 5.164.

Explanation

The first step is to find the square root of 5/3, which is approximately 1.29.

The second step is to multiply 1.29 by 4. 1.29 × 4 = 5.164.

Well explained 👍

Problem 4

What will be the square root of (9 + 1/3)?

Okay, lets begin

The square root is approximately 3.055.

Explanation

To find the square root, we need to find the sum of (9 + 1/3). 9 + 1/3 = 28/3, and then √(28/3) ≈ 3.055.

Therefore, the square root of (9 + 1/3) is approximately ±3.055.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length 'l' is √(5/3) units and the width 'w' is 3 units.

Okay, lets begin

We find the perimeter of the rectangle as approximately 8.58 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√(5/3) + 3).

Perimeter ≈ 2 × (1.29 + 3) = 2 × 4.29 ≈ 8.58 units.

Well explained 👍

FAQ on Square Root of 5/3

1.What is √(5/3) in its simplest form?

The simplest form of √(5/3) is √5/√3. This can be further rationalized to (√15)/3.

2.Mention the factors of 5/3.

3.Calculate the square of 5/3.

We get the square of 5/3 by multiplying the fraction by itself, that is (5/3) × (5/3) = 25/9.

4.Is 5/3 a prime number?

5.5/3 is divisible by?

As a fraction, 5/3 is not divisible by integers in the same way whole numbers are. It is already in its simplest form.

Important Glossaries for the Square Root of 5/3

  • Square root: A square root is the inverse of a square. Example: 4^2 = 16, and the inverse of the square is the square root, that is √16 = 4.
  • Irrational number: An irrational number cannot be written in the form of p/q, where q is not equal to zero and p and q are integers. √(5/3) is an example.
  • Principal square root: Though a number has both positive and negative square roots, the positive square root is often used for practical purposes. This is known as the principal square root.
  • Radical: A radical is a symbol (√) used to denote the square root or nth root of a number. For instance, √(5/3) represents the square root of 5/3.
  • Rationalization: The process of eliminating a radical from the denominator of a fraction by multiplying the numerator and the denominator by a suitable value is called rationalization.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.