Introduction
In fact, the square root generates a value that, when multiplied by itself, yields the initial number. As a result, it is the original number’s root.
For example: To find the square of a number 9, Using the multiplication operation \(9\times 9=81\), we can multiply a number by itself to find its square.
Similarly, we will perform the inverse operation to determine the square root of 81. The radical symbol for a square root is: ‘ \(\sqrt{~~}\) ’.
Since, \(9^2=81\)
\(\Rightarrow\sqrt{81}=\sqrt{9^2}=9\)
What is the Value of Root 3?
Three is an irrational number, as is its square root. In honor of Theodorus of Cyrene, who demonstrated its irrationality, it is also known as Theodorus constant. When root 3 is multiplied by itself, the result is 3, which is a positive real number. The square root of three can be expressed in various ways.
- In decimal form it is expressed as 1.732
- In radical form it is expressed as \(\sqrt3\)
- In exponent form it is expressed as \(3^\frac{1}{2}\)
How to find the Square Root of 3?
Without using a calculator, we must first determine whether the real number under the root is a perfect square before we can calculate its square root. The prime factorization method makes it simple to determine the square root of a number if it is a perfect square. But we must use the long division technique to determine the root of non-perfect squares.
For example, Since \(3^2=9\), the square root of 9 is \(\pm3\). However, the number 3 in this instance is a non-perfect square. Thus, we are unable to apply the factorisation method. As an alternative, we will calculate the square root of 3 using the long division method.
Deriving the Square Root 3
We can apply the following steps to determine the square root of 3 using the long division method.
Step 1: Rewrite the number as shown below.
\(\overline{3}\ .\ \ \overline{00}\ \ \overline{00}\ \overline{00}\)
Step 2: Take a number whose square is less than or equal to 3.
\(1^2=1\), which is less than 3. So we will take 1.
Step 3: Write the number 1 as the divisor and 3 as the dividend. Now divide 3 by 2.
Here, Quotient = 1 and Remainder = 2.
Step 4: Bring down 00 and write it after 2, so the new dividend is 200 and add the quotient 1 to the divisor, that is, 1 + 1 = 2.
Step 5: Add a digit right next to 2 to get a new divisor such that the product of a number with the new divisor is less than or equal to 200.
27 x 7 = 189, which is less than 200.
Subtract the product 189 from 200 to get the remainder.
Here, Quotient = 1.7 and Remainder = 11
Step 6: Repeat the previous two steps to obtain the quotient up to three decimal places.
Therefore, the value of the square root of 3, that is, \(\sqrt3\) is approximately 1.732.
Positive and Negative Square Roots of a Number
Each number has two square roots, one is positive and other is negative. Let us understand this with an example, multiply \(-\sqrt3\) by itself.
\((-\sqrt3)\times(-\sqrt3)=\sqrt{3^2}=3\) [Product of two negatives is a positive]
Also, multiply \(\sqrt3\) by itself
\(\sqrt3\times\sqrt3=\sqrt{3^2}=3\)
From these two operations the square of \(-\sqrt3\) is 3, and the square of \(\sqrt3\) is also 3. So we can say that the square roots of 3 are both \(\sqrt3\) and \(-\sqrt3\).
In general,
- \(\sqrt y\) represents the positive square root of \(y.\)
- \(-\sqrt y\) represents the negative square root of \(y.\)
- \(\pm\sqrt y\) represents both the square roots of \(y.\)
Perfect Squares and Imperfect Squares
The square root of a perfect square number is always an integer. On the other hand, the value of the square root of imperfect squares is a non-integer, that is, it contains decimals or fractions.
For example, 12 is the square root of 144, which is a perfect square, and 3.464 is the square root of 12, which is an imperfect square.
Rapid Recall
Note: This table lists the approximate values of the square roots of some numbers that can be memorized to determine the square roots of higher imperfect square numbers.
Number | Square root |
|---|---|
2 | 1.414 |
3 | 1.732 |
5 | 2.236 |
7 | 2.646 |
Solved Examples
Example 1: Find the result of multiplying 15 by \(\sqrt3.\)
Solution:
We have to find the value of \(15\times\sqrt3\)
Since the value of \(\sqrt3=1.732\)
\(15\ \times\ \sqrt3=15\times1.732\) [Multiply]
= 25.98
Therefore, the value of 15 multiplied by \(\sqrt3\) is 25.98.
Example 2:Find the result of \(\left(6\sqrt3\ \right)+12\left(\sqrt3\times\sqrt3\ \right).\)
Solution:
Given : \(\left(6\sqrt3\ \right)+12\left(\sqrt3\times\sqrt3\ \right)\)
\(\left(6\sqrt3\ \right)+12\left(\sqrt3\times\sqrt3\ \right) =\left(6\sqrt3\ \right)+12\ \left(\sqrt3\times\sqrt3\right)\) (By using the PEMDAS rule]
= \(\left(6\times1.732\ \right)+12\left(\sqrt{3}\ \right)^2\) (As we know, \(\sqrt3\) = 1.732)
= \(\left(10.392\right)+\left(12\times3\right)\) (Simplify)
= 10.392 + 36
= 46.392
Thereby, the value of the given expression
\(\left(6\sqrt3\ \right)+12\left(\sqrt3\times\sqrt3\ \right)\) is 46.392.
Example 3: For precisely three hours, Julie is moving down the highway at an average speed of 10\(\sqrt3\) km/h. How far does she travel in one day?
Solution:
We need to use the formula, Distance = Speed x Time
Speed = \(10\sqrt3\) = 17.320 km/hr (As we know, \(\sqrt3\) = 1.732)
Time = 3 hr
Using the formula, Distance = 17.320 x 3 = 51.96
Thereby, Julie traveled a distance of 51.96 kilometers.
There are two square roots of 3 which are 1.732 and -1.732.
No, 3 is not a perfect square because its square root, expressed in decimal form, is 1.732 is an irrational number.
The square root of 3 = 1.732..
In light of the fact that it is a non-terminating and non-repeating value, the square root of 3 is an irrational number.