What is Square Root of 13?
We already know that 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 13 is written as \(\sqrt{13}\), with the radical sign ‘ \(\sqrt{}\) ‘ and the radicand being 13. The square root of 13 has a value that is nearly equal to 3.60555……, and this value is a non-terminating and a non-repeating value, showing us that it is an irrational number.
How did we obtain the value of \(\sqrt{13}\) as 3.60555…? We use the long division method to find this value.
Deriving the Square Root of 13
We can apply the following steps to determine the square root of 13 using the long division method.
Step 1: Rewrite the number as shown below.
\(\overline{13}\). \(\overline{00}\) \(\overline{00}\) \(\overline{00}\)
Step 2: Take a number whose square is less than or equal to 13.
\(3^2\) = 9, which is less than 13. So we will take 3.
Step 3: Write the number 3 as the divisor and 13 as the dividend. Now divide 13 by 3.
Here, Quotient = 3 and Remainder = 4.
Step 4: Bring down 00 and write it after 4, so the new dividend is 400 and add the quotient 3 to the divisor, that is, 3 + 3 = 6.
Step 5: Add a digit right next to 6 to get a new divisor such that the product of a number with the new divisor is less than or equal to 400.
66 \(\times\) 6 = 396, which is less than 400.
Subtract the product 396 from 400 to get the remainder.
Here, Quotient = 3.6 and Remainder = 4
Step 6: Repeat the previous two steps to obtain the quotient up to three decimal places.
Therefore, the value of the square root of 13, that is, 13 is approximately 3.605.
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 \(-\sqrt{13}\) by itself.
(\(-\sqrt{13}\)) \(\times\) (\(-\sqrt{13})=13\) [Product of two negatives is a positive]
Also, multiply \(\sqrt{13}\) by itself
\(\sqrt{13}\times\sqrt{13}=13\)
From the above the square of \(-\sqrt{13}\) is 13 and square of \(\sqrt{13}\) is also 13. This leads to the fact that square roots of 13 are both \(\sqrt{13}\) and \(-\sqrt{13}\).
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 3.605 is the square root of 13, 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 larger imperfect square numbers.
Solved Square Root of 13 Examples
Example 1: Find the square root of 13 by the approximation of the prime number square roots:
Solution:
Find the prime factorization of the number 13.
13 = 13 \(\times\) 1
\(\sqrt{13}=\sqrt{13\times 1}=\sqrt{13}\times\sqrt{1}\) [Square root of 1 is \(\sqrt{1}\) = 1]
= 3.605 [Square root of 13 is \(\sqrt{13}\) ≈ 3.605]
Example 2: Find the Square root of 78
Find prime factorization of 78 as below
Solution:
\(78=2\times 3 \times 13\) Write the equation
\(\sqrt{78}=\sqrt{2\times 3 \times 13}\) Apply square root both side
\(\sqrt{78}=\sqrt{2}\times\sqrt{3} \times\sqrt{13}\) Use property \(\sqrt{abc}=\sqrt{a}.\sqrt{b}.\sqrt{c}\)
\(\sqrt{78}=1.414\times 1.732 \times 3.615\) Substitute 1.414 for \(\sqrt{2}\) and 1.732 for \(\sqrt{3}\) and 3.615 for \(\sqrt{13}\)
\(\sqrt{78}\approx 8.853\) Simplify
Hence, the square root of 78 is 8.853.
Example 3 : Find the length of the boundary of a circular shaped garden with radius as 20\(\sqrt{13}\) feet.
Solution:
The length of the boundary of a circle is called circumference.
C = 2\(\pi\)r Write the formula for circumference
C = 2 \(\times\) 3.14 \(\times\) 20\(\sqrt{13}\) Substitute 3.14 for \(\pi\) and 20\(\sqrt{13}\) for r
C = 125.6 \(\times\) 3.615 Simplify and substitute 3.615 for \(\sqrt{13}\)
C = 454.0444 \(\approx\) 54 Simplify
So, the circumference of the garden is approximately 454 feet.
A number is said to be a perfect square if the square root of the number is an integer.
144 is a perfect square because the square root of 144 is 12.
81 is a perfect square as the square root of 81 is 9.
Yes , the product of two perfect squares is always a perfect square.
Example: Let us have two perfect square numbers, 25 and 64. The product of 24 and 64 is 1600, which is also a perfect square with its square root being 40.
The measure of length is always positive. So the negative square of a number root is neglected. So we use 13 as the measure of the side length of a square field.
The square root of a negative number is imaginary. So by using the concept of complex numbers we can find the square root of a negative number.