Square Root of 75 Explained with Simplified Radical Form

The concept of square root of 75 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. It is a commonly used topic for simplifying radical expressions and calculating approximate values.

Understanding Square Root of 75

A square root of 75 is a value which, when multiplied by itself, gives 75 as a result. In mathematical notation, this is written as √75. This concept is widely used in finding radical form, decimal values, and in simplification steps for surds in algebra and geometry.

Formula Used in Square Root of 75

The standard formula is: \( \sqrt{75} \)

To find the value, you can use prime factorization:
\( \sqrt{75} = \sqrt{3 \times 5 \times 5} = 5\sqrt{3} \)

How to Simplify Square Root of 75 (Step-by-Step)

Let’s break down the simplification of √75 to its simplest radical form:

So, the square root of 75 simplified is 5 times the square root of 3.

Square Root of 75 in Decimal Form

The decimal value of the square root of 75 (rounded to three decimal places) is:

This value is useful for calculator-based questions and quick approximations in exams.

Radical and Fractional Form of Square Root of 75

Radical form: √75
Simplest radical form: 5√3
Decimal form: 8.660
Fractional/approximate rational form: There is no exact rational fraction for √75, but you can approximate using fractions if required (e.g., 8⅔).

Comparison With Nearby Square Roots

See how √75 compares with close values:

This table shows how the square root of 75 fits between 20 and 76, helping you compare patterns in radical simplification.

Square Root of 75 by Long Division Method

To find √75 manually up to three decimals, use these steps:

This long division method is used in exams for manual calculation of square roots.

Worked Example – Solving a Problem

Let’s solve a typical problem involving the square root of 75:

You can also use the simplified radical form, if required: 20 + 5√3.

Cube Root of 75

For completeness, the cube root of 75 is expressed as:

Cube roots are solved separately using their own procedures and help differentiate from square roots in advanced problems.
For more on cube roots, see Cube Root of Numbers.

Common Mistakes to Avoid

• Confusing square root of 75 with a perfect square (it is NOT a perfect square, as 8²=64, 9²=81).
• Forgetting to simplify √75 into 5√3 in radical questions.
• Writing square root answers as fractions without rationalizing or approximating correctly.

Real-World Applications & Writing in Words

The square root of 75 appears in geometry (finding diagonal lengths), physics (measuring distances), and engineering calculations. In words, write as: “square root of seventy-five.”
An example: “Find the side of a square with area 75 square units.” The answer would be √75 units, simplified as 5√3 units.

Page Summary

We learned what the square root of 75 is, its simplified radical and decimal forms, how to calculate it step by step, and how it is used in real-world situations. For more maths concepts and practice, use Vedantu’s resources and internal links for in-depth learning.

Further Learning & Suggested Links

Explore related topics to master roots, surds, and mathematical simplification:

• Square Root of 20 – Compare and learn simplification patterns with other non-perfect squares.
• Factors of 75 – Understand prime factorization for root calculation.
• Square Root Finder – Practice and check other square roots quickly.
• Square Root Tricks – Discover quick methods for solving square roots.
• Square Root Table – Reference other roots for exams and practice.
• Cube Root of Numbers – Avoid confusion between square and cube roots.
• Surds – Learn the properties and simplification of irrational roots like √75.
• Multiples of 4 – Strengthen understanding of factors and multiples, key for simplification.
• Tables 2 to 20 – Master multiplication for fast calculations.
• Rational and Irrational Numbers – See why √75 is irrational and the meaning of such roots.
• Basic Geometrical Ideas – Find out where square roots apply in geometry.