What is Square Root of 3?
The square root of a number is a value that, when multiplied by itself, gives us the original number. Hence, finding the square root is the converse of finding the square of a number.
The square root of \( 3 \) is written as \( \sqrt{3} \), with the radical sign ‘\( \sqrt{} \)‘ and the radicand being \( 3 \). The square root of \( 3 \) has a value that is nearly equal to \( 1.7320508.. \), and this value is non-terminating and non-repeating, showing us that it is an irrational number.
How did we obtain the value of \( \sqrt{3} \) as \( 1.7320508…? \) We use the long division method to find this value.
Deriving the Square Root of 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 \( 200 \).
\( 27\times 7=189 \), 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, \( \sqrt{3} \) is approximately \( 1.732 \).
Positive and Negative Square Root of a Number
Each number has two square roots, one is positive and other is negative. Let us understand this with an example, multiply \( -\sqrt{3} \) by itself.
\( (-\sqrt{3})\times (-\sqrt{3})=(\sqrt{3})^2=3 \) [Product of two negatives is a positive]
Also, multiply \( \sqrt{3} \) by itself
\( \sqrt{3}\times \sqrt{3}=\sqrt{3^2}=3 \)
From the above the square of \( -\sqrt{3} \) is \( 3 \) and square of \( \sqrt{3} \) is also \( 3 \), so we can say that square roots of \( 3 \) are both \( \sqrt{3} \) and \( -\sqrt{3} \).
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, \( 10 \) is the square root of \( 100 \), which is a perfect square, and \( 3.1622 \) is the square root of \( 10 \), 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.
Solved Square Root of 3 Examples
Example 1: Find the square root of \( 3 \) by the approximation of the prime number square roots:
Solution:
Find the prime factorization of the number \( 3 \).
\( 3=3\times 1 \)
\( \sqrt{3}=\sqrt{3\times 1} \) [Apply square root both side]
\( \Rightarrow \sqrt{3}\times \sqrt{1} \) [Use property \( \sqrt{ab}=\sqrt{a}\cdot \sqrt{b} \)]
\( \Rightarrow 1.732\times 1 \) [Substitute \( 1 \) for \( \sqrt{1} \) and \( 1.732 \) for \( \sqrt{3} \)]
\( \Rightarrow 1.732 \) [Multiply]
So, the value of the square root of \( 3 \) is \( 1.732 \).
Example 2: Find the square root of \( 12 \).
Solution:
Use the factor tree method to find the prime factors of \( 12 \).
\( 12=2\times 2\times 3 \)
\( \sqrt{12}=\sqrt{2\times 2\times 3} \) [Apply square root both side]
\( \sqrt{12}=\sqrt{2^2\times 3} \) [Write \( 2\times 2 \) as \( 2^2 \)]
\( \sqrt{12}=\sqrt{2^2}\times \sqrt{3} \) [Use property \( \sqrt{ab}=\sqrt{a}\cdot \sqrt{b} \)]
\( \sqrt{12}=2\times 1.732 \) [Substitute \( 2 \) for \( \sqrt{2^2} \) and \( 1.732 \) for \( \sqrt{3} \)]
\( \sqrt{12}=3.464 \) [Multiply]
Example 3: John is riding a bicycle at speed of \( 5\sqrt{3} \) miles per hour. What is the distance he covers in \( 1.5 \) hours?
Solution:
Use the speed formula to find the covered distance.
\( \text{Speed}=\frac{\text{Distance}}{\text{Time}} \)
\( \text{Distance}=\text{Speed}\times \text{Time} \) [Rearrange the formula for distance]
\( \text{Distance}=5\sqrt{3}\times 1.5 \) [Substitute \( 5\sqrt{3} \) for speed and \( 1.5 \) for time]
\( \text{Distance}=5\times 1.732\times 1.5 \) [Substitute \( 1.732 \) for \( \sqrt{3} \)]
\( \text{Distance}=12.99\approx 13 \) [Multiply]
So, John covers almost \( 13 \) miles in \( 1.5 \) hours.
No, the square root of numbers may and may not be rational. Square roots of non-perfect square numbers are irrational numbers. For example, square root of \( 7 \) i.e.\( \sqrt{7} \),which is not rational, that is, an irrational number. However, the square roots of a perfect square number are rational. For example, square root of \( 9 \), \( \sqrt{9}=\pm 3 \), which is a rational number.
Yes, the square root of any decimal number can be calculated by the method called long division method.
Yes, the product of two perfect squares is always a perfect square.
Example: Let’s have two perfect square numbers, \( 25 \) and \( 64 \). When we multiply \( 25 \) and \( 64 \) we get \( 1600 \) as the product, and the square root of \( 1600 \) is \( 40 \).
A number is said to be a perfect square if the square root of the number is an integer. For example: \( 169 \) is a perfect square because the square root of \( 169 \) is \( 13 \), which is an integer.