How to Simplify the Square Root of 48 with Examples
The concept of square root of 48 is essential in mathematics and helps in solving real-world and exam-level problems efficiently. Understanding how to simplify the square root of 48 helps students tackle homework, board exam questions, and competitive maths problems with confidence.
Understanding Square Root of 48
A square root of a number is a value that, when multiplied by itself, gives the original number. The square root of 48 is written as √48. This concept appears in simplifying roots, solving quadratic equations, geometry, and surds. Square roots are especially useful in areas such as area calculations, Pythagoras theorem applications, and algebraic manipulation.
Value and Notation of Square Root of 48
The square root of 48 can be written in both radical and decimal form:
4√3 (simplest radical form)
6.928 (decimal, rounded to three decimal places)
This means \( \sqrt{48} = 4\sqrt{3} \approx 6.928 \).
Here’s a helpful table to understand the square root of 48 and its comparison to nearby values:
Square Root of 48 Table
| Form | Value | Perfect Square? |
|---|---|---|
| √48 (radical form) | 4√3 | No |
| √49 | 7 | Yes |
| √36 | 6 | Yes |
| √48 (decimal approx) | 6.928 | No |
This table helps you see that the square root of 48 is not a perfect square, and how its value fits between perfect squares.
How to Simplify Square Root of 48: Step-by-Step
Let's break down the simplification of the square root of 48 step by step for clarity:
2. Find its prime factors:
3. Pair the similar factors under the root:
4. Take pairs out of the square root:
5. So, you get:
6. Using √3 ≈ 1.732,
Final Answer:
√48 = 4√3 ≈ 6.928
Is 48 a Perfect Square?
No, 48 is not a perfect square because there is no integer that when multiplied by itself gives exactly 48. Its square root cannot be expressed as a whole number. Instead, the square root of 48 simplifies to a surd, which is 4√3.
Worked Example – Square Root Using the Division Method
Let’s find the square root of 48 using the long division method step-by-step:
2. The closest square less than or equal to 48 is 6×6=36.
3. Bring down the next pair of zeros to make 1200.
4. 129×9=1161 (which fits).
5. Bring down next pair of zeros (3900). Double previous result (69), get 138.
6. Continue the process for higher decimal accuracy.
7. After three decimal places, we get the answer: 6.928
You can practice all steps for other numbers using the full method on the page Square Root by Division Method.
Practice Problems
- Simplify the square root of 12 in radical form.
- Is the square root of 49 a whole number? Why?
- Find the approximate value of square root of 11 using step-by-step division.
- Write 48 as a product of its prime factors and verify the radical simplification.
Common Mistakes to Avoid
- Stopping after writing √48 = 4√3 but not calculating the decimal value for checking.
- Forgetting to pair the factors inside the radical correctly.
- Confusing square roots with cube roots—remember, √48 ≠ ∛48.
Real-World Applications
You may encounter the square root of 48 in real life when working with square areas (such as finding side length if area is 48 square units), or while using the Pythagoras theorem in geometry problems. The concept also appears in algebra, physics, and architectural planning. Vedantu helps students connect such maths concepts to real-world situations through examples and practice problems.
We explored the idea of the square root of 48, saw how to simplify it, found its exact radical and approximate decimal forms, and reviewed stepwise solutions. Keep practising these simplifications and methods with Vedantu, and use stepwise logic whenever you work with square roots in school or exams.
Explore Related Topics
- Factors of 48 – Understand factorization and apply it to root simplifications.
- Square Root by Division Method – Learn the stepwise method for all non-perfect squares.
- Square Root Finder – Find and verify roots quickly.
- Square Root Table – Compare square root values across numbers.
- Square Root of 49 – See how perfect squares work, and contrast with 48.
- Factors of 12 – Use factorization ideas for radical simplification.
- Cube Root of 48 – Avoid confusion with cube roots and see the difference.
- Value of Root 2 – Useful when comparing surd and decimal forms.
- Surds – Learn how to handle and simplify irrational roots.
FAQs on Square Root of 48: Easy Steps, Formula & Decimal Value
1. What is the square root of 48?
The square root of 48 is a number which, when multiplied by itself, equals 48. It is symbolically written as √48, and its value is approximately 6.928 when rounded to three decimal places.
2. How is the square root of 48 simplified?
To simplify √48, express 48 as the product of its prime factors: 48 = 2 × 2 × 2 × 2 × 3. Grouping pairs of prime factors and taking them outside the square root gives √48 = 4√3, which is the simplest radical form.
3. What is the simplest radical form of √48?
The simplest radical form of √48 is 4√3. This is obtained by prime factorizing 48 and simplifying by taking out perfect square factors, leading to a cleaner and more exam-friendly expression.
4. Is 48 a perfect square number?
No, 48 is not a perfect square number because it cannot be expressed as the product of two equal integers. This is why its square root results in an irrational number and requires simplification into radical form.
5. How do you calculate √48 by division method?
To calculate √48 using the long division method, follow these steps:
1. Write 48 as 48.000000 and pair the digits from right to left.
2. Find the largest number whose square is less than or equal to 48 (which is 6).
3. Perform successive division and subtraction steps by doubling the quotient and finding suitable digits.
4. Continue the process to get the decimal approximation up to three decimal places, resulting in approximately 6.928.
6. Why is √48 often confused with √49 in exams?
√48 is often confused with √49 because 49 is a perfect square with an exact root of 7, which is a whole number, while 48 is not. This close numerical proximity can cause students to mistakenly write √48 as 7 or approximate it incorrectly instead of using its simplified radical form or decimal approximation.
7. How does knowing prime factors help in simplifying √48?
Understanding the prime factorization of 48 helps simplify √48 by identifying pairs of prime factors that can be taken outside the square root. This method breaks down complex roots into simpler forms, making calculations easier and improving accuracy during exams.
8. Can √48 be used in area or geometry questions?
Yes, √48 can appear in geometry problems, especially in questions involving lengths, diagonals, or areas where exact irrational values are needed. Using the simplified form 4√3 or its decimal approximation helps solve such problems accurately.
9. Why do some students write √48 as 6.9 in answers?
Some students approximate √48 as 6.9 for simplicity, but this is a rounded value and less accurate than the correct approximation (~6.928). While rounding is sometimes acceptable, in board exams and precise calculations, the simplified radical form 4√3 or a more accurate decimal is preferred.
10. What common mistakes are made during board exam simplification steps?
Common mistakes include:
• Failing to prime factorize correctly.
• Not grouping prime factors into pairs.
• Incorrectly simplifying the radical form.
• Mixing up perfect square factors and leftovers.
• Rounding decimal approximations prematurely.
To avoid these, carefully follow step-by-step factorization and simplification methods.
11. What is the value of √48 in decimal form?
The decimal value of √48 is approximately 6.928, rounded to three decimal places. This helps when a numerical approximation is necessary for calculations or comparisons.
12. How is the square root of 48 written in words?
The square root of 48 is written as "the square root of forty-eight" or verbally expressed as "root forty-eight". Knowing this helps students articulate and understand problems clearly in exams.
