How to Simplify and Calculate the Square Root of 75
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:
2. Take out pairs of the same number (since √(a²) = a): The pair is two 5's.
3. Move one 5 outside the root: √75 = √(5×5×3) = 5√3.
4. Final simplified radical form: √75 = 5√3
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:
| Number | Square Root Value | Simplest Radical Form |
|---|---|---|
| 20 | 4.472 | 2√5 |
| 75 | 8.660 | 5√3 |
| 76 | 8.718 | irrational |
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:
2. Find the largest number whose square is less than or equal to 75 (8×8=64). Subtract 64 from 75 to get 11.
3. Bring down two zeros: new dividend is 1100.
4. Double the root so far (8×2=16); determine the largest digit X such that (160+X)×X ≤ 1100. X=6 (166×6=996). Subtract, continue.
5. Repeat the process for more decimal points (see detailed method above).
6. The answer is 8.660.
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:
Step 1: Use the decimal value √75 ≈ 8.660.
Step 2: Add to 20: 20 + 8.660 = 28.660.
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.
FAQs on Square Root of 75: Step-by-Step Guide, Value & Simplest Radical Form
1. What is 75 square root simplified?
The square root of 75 simplified is expressed in its simplest radical form as 5√3. This is found by prime factorization of 75 = 3 × 5 × 5, and then simplifying the square root by taking out pairs.
2. How to find √75 step by step?
To find √75 step by step, you can use the prime factorization method: (1) Factor 75 into primes: 3 × 5 × 5, (2) Pair the 5s and take one 5 outside the root, (3) Resulting in 5√3. Additionally, you can use the long division method to calculate the decimal value approximately as 8.66.
3. What is the decimal value of the square root of 75?
The decimal value of √75 is approximately 8.660, which can be rounded to 8.66. This value is useful for calculations and can be obtained either using a calculator or manually through the long division method.
4. How do you write the square root of 75 in radical form?
The radical form of the square root of 75 is written as 5√3. This comes from simplifying the root by extracting the pair of 5s from the square root of 75 = 3 × 5 × 5, which results in 5 times the square root of 3.
5. What is the cube root of 75?
The cube root of 75 is the value that when multiplied by itself three times equals 75. It is approximately 4.217. Unlike the square root, the cube root is not simplified in radical form for 75 as it is not a perfect cube.
6. Why is 75 not a perfect square?
The number 75 is not a perfect square because it cannot be expressed as the product of an integer multiplied by itself. Its prime factors (3 and 5²) do not form perfect pairs for all primes, leaving an irrational square root.
7. Why do some students confuse the square root of 75 with the square root of 76?
Students sometimes confuse √75 with √76 because the two numbers are close in value, and their roots are approximately 8.66 and 8.72 respectively. Careful simplification and decimal calculation help differentiate these values clearly.
8. What mistakes happen when simplifying √75 in exams?
Common mistakes when simplifying √75 include: (1) forgetting to factor 75 into primes correctly, (2) leaving the radical unsimplified, (3) confusing it with √(25×3) and missing to take out the 5 properly, (4) incorrect decimal rounding without sufficient precision.
9. How does √75 relate to multiplication tables for 75?
Understanding √75 connects to the multiplication tables of 3 and 5, since 75 = 3 × 25 and 25 is 5 squared. Recognizing these tables helps in prime factorization and simplifying square roots effectively.
10. Why is knowing both radical and decimal forms important for board exams?
Knowing both the radical form and decimal value of √75 is important because exams may require either form depending on the question. Radical form shows the simplest exact expression, while decimal form aids numerical calculations and estimations.
11. What is the difference between radical form and decimal form of √75?
The radical form of √75 is 5√3, which is an exact representation using square root notation. The decimal form is an approximate numeric value of 8.66 derived from calculating the square root value to a certain number of decimal places.
12. Is the square root of 75 a rational or irrational number?
The square root of 75 is an irrational number because it cannot be expressed as a ratio of two integers (p/q). Its decimal form is a non-terminating, non-repeating decimal, and it remains in radical form for exactness.
