How to Calculate and Simplify the Square Root of 6
The square root is a topic that many students find difficult to understand. However, once you understand the concept well, finding square roots will no longer be a challenge for you. So, in simple words square root, in mathematics, is a factor of a number that, when multiplied by itself, gives the original number. For example, both 3 and –3 are square roots of 9.
The square root of a number a is the number b such that b² = a. The square root of any number is represented by the symbol and is also often known as radical. The number or expression given under the square root symbol is known as radicand. The square root is a commonly used function in Mathematics. It is widely used in subjects like Mathematics and Physics. Sometimes it is tedious to find the square root of a number, especially the numbers which are not the perfect square of the number. In this article, we will discuss the square root of 6, and how to calculate the root 6 value using the simplifying square root method.
What is the root 6 value?
The root 6 value is 2.449
Square Root of 6 Definition
The square root of a number 6 is a number y such that y² = 6. The square root of 6 in radical form is written as √6. The square root of 6 in radical form is expressed as√6
How to Calculate the Under Root 6 Value?
We can calculate the under root 6 value using different methods of square roots. These methods can be the long division methods, prime factorization method, or simplifying square root method. Let us discuss how to calculate the under root 6 value using the simplifying square root method.
To simplify a square root, make the number under the square root as small as possible while keeping it as a whole number. Mathematically, it can be expressed as: √x.y=√x×√y
To express the square root of 6 in the simplest form, we will make the number 6 as small as possible, ensuring to keep it as a whole number. Hence, the square root of 6 in simplest form is represented as√6=√2×√3. This can be further simplified by substituting the value of √2 and √3
\[\sqrt{6}=\sqrt{2}\times \sqrt{3}\]
\[\sqrt{6}=1.414\times 1.732\]
\[\sqrt{6}=2.449\]
Hence, the square root of 6 in simplest form is 2.449. Similarly, we can also calculate the square root of any other whole numbers and their factors. Hence, simplifying the square root method is the simplest method of calculating the square root. We can also calculate the value of under root 6 using the calculator as this will give us the exact value. The exact value of the square root will always be given in a decimal number as it is impossible to determine a positive whole integer as a root for non-rational numbers.
Simplifying the Square Root Using Perfect Square Method
Following are the steps to simplify the square root using the perfect square method:
Find the perfect square that divides the number in the radicand.
Express the numbers as a factor of a perfect square.
Simplify the radicals.
Solved Example
1. Simplify $\mathbf{\sqrt{300}}$
Solution:
\[\sqrt{300}=\sqrt{100\times 3}\]
\[\sqrt{300}=\sqrt{10\times 10\times 3}\]
\[\sqrt{300}=10\sqrt{3}\]
Hence,\[\sqrt{300}\] can be simplified as \[10\sqrt{3}\]
2. Simplify the following radical expressions :
\[\sqrt{48}\]
\[\sqrt{75}\]
Solutions:
i.$\mathbf{\sqrt{48}}$
Step 1: The perfect square 16 will divide the number 48.
Step 2: Express 48 as a factor of 16
48=16×3
Step 3: Reduce the square root of 16 as shown below:
\[\sqrt{48}=\sqrt{16\times 3}\]
\[\sqrt{48}=\sqrt{4\times 4\times 3}\]
\[\sqrt{48}=4\sqrt{3}\]
Hence,\[\sqrt{48}\] can be simplified as\[4\sqrt{3}\]
ii.$\mathbf{\sqrt{75}}$
Step 1: The perfect square 25 will divide the number 75.
Step 2: Express 75 as a factor of 25.
75=25×3
Step 3: Simplify the radicals as shown below:
\[\sqrt{75}=\sqrt{25\times 3}\]
\[\sqrt{75}=\sqrt{5\times 5\times 3}\]
\[\sqrt{75}=5\sqrt{3}\]
Hence,\[\sqrt{75}\] can be simplified as\[5\sqrt{3}\]
FAQs on Square Root of 6 Explained
1. What is the approximate value of the square root of 6?
The value of the square root of 6 (√6) is an irrational number, meaning its decimal representation never ends and never repeats. For practical calculations, it is often approximated to a few decimal places. The value of √6 is approximately 2.449. This value is found using methods like the long division method.
2. Why is the square root of 6 considered an irrational number?
The square root of 6 is an irrational number because 6 is not a perfect square. A perfect square is a number that is the product of an integer with itself (e.g., 4 = 2x2, 9 = 3x3). Since there is no integer that, when multiplied by itself, equals 6, its square root cannot be expressed as a simple fraction (p/q). This results in a non-terminating, non-repeating decimal, which is the definition of an irrational number.
3. How can the square root of 6 be expressed in its simplest radical form?
To simplify a radical, we look for perfect square factors in the number under the root. The prime factorization of 6 is 2 × 3. Since neither 2 nor 3 are perfect squares, the square root of 6 cannot be broken down further. Therefore, √6 is already in its simplest radical form.
4. How do you find the square root of 6 using the long division method?
The long division method provides a step-by-step way to approximate the square root of 6. Here is the process:
Step 1: Find the largest integer whose square is less than or equal to 6. This is 2, as 2² = 4.
Step 2: Place 2 as the first digit of the answer. Subtract 4 from 6, leaving a remainder of 2.
Step 3: Add a decimal point and bring down a pair of zeros, making the new dividend 200.
Step 4: Double the current quotient (2), which gives 4. Find a digit 'x' such that 4x × x is close to 200. This digit is 4 (since 44 × 4 = 176).
Step 5: Place 4 as the next digit in the answer (making it 2.4). Subtract 176 from 200 to get 24.
Step 6: Repeat the process by bringing down another pair of zeros. This iterative process can be continued to find the value to any desired decimal place.
5. How does the square root of 6 compare to the square roots of perfect squares like 4 and 9?
Comparing with perfect squares helps in estimating the value of √6. We know that the square root of 4 is 2 (√4 = 2) and the square root of 9 is 3 (√9 = 3). Since the number 6 lies between the two perfect squares 4 and 9, its square root must lie between their respective roots, 2 and 3. This confirms that the value of √6 is greater than 2 but less than 3, which is consistent with its approximate value of 2.449.
6. Since 6 is an even number, why isn't its square root a simple whole number?
Whether a number is even or odd does not determine if its square root is a whole number. For a number to have a whole number square root, it must be a perfect square. An even number like 6 is not a perfect square because no integer multiplied by itself equals 6. For instance, other even numbers like 4 (2x2) and 36 (6x6) are perfect squares and have whole number roots, whereas even numbers like 2, 6, 8, and 10 are not.
7. What is the difference between the square of 6 and the square root of 6?
The square of 6 and the square root of 6 are two distinct mathematical operations.
The square of 6 means multiplying 6 by itself, which is 6 × 6 = 36.
The square root of 6 (√6) is a number that, when multiplied by itself, gives 6. As we know, this value is approximately 2.449.
Essentially, one operation (squaring) makes the number larger, while the other (finding the square root) makes it smaller.
