What is Square Root of 5
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 5 is written as \(\sqrt{5}\), with the radical sign ‘ \(\sqrt{}\) ‘and the radicand being 5. The square root of 5 has a value that is nearly equal to 2.2306797, and this value is non-terminating and non-repeating, showing us that it is an irrational number.
How did we obtain the value of \(\sqrt{5}\) as 2.2306797 ? We use the long division method to find this value.
Deriving the Square Root 5
We can apply the following steps to determine the square root of 5 using the long division method.
Step 1: Rewrite the number as shown below.
\(\overline{5}\). \(\overline{00}\) \(\overline{00}\) \(\overline{00}\)
Step 2: Take a number whose square is less than or equal to 5.
\(2^2\) = 4, which is less than 5. So we will take 2.
Step 3: Write the number 2 as the divisor and 5 as the dividend. Now divide 5 by 2.
Here, Quotient = 2 and Remainder = 1.
Step 4: Bring down 00 and write it after 1, so the new dividend is 100 and add the quotient 2 to the divisor, that is, 2 + 2 = 4.
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 100.
42 \(\times\) 2 = 84, which is less than 100.
Subtract the product 84 from 100 to get the remainder.
Here, Quotient = 2.2 and Remainder = 16
Step 6: Repeat the previous two steps to obtain the quotient up to three decimal places.
Therefore, the value of the square root of 5, that is, \(\sqrt{5}\) is approximately 2.236.
Positive and Negative Square Roots of a Number
Each number has two square roots, one is positive, and the other is negative. Let us understand this with an example. When we multiply -\(\sqrt{5}\) by itself we see that the result is 5.
Hence, (-\(\sqrt{5}\)) \(\times\) (-\(\sqrt{5}\)) = \(\sqrt{5^2}\) = 5 [Product of two negatives is a positive]
Also, multiply \(\sqrt{5}\) by itself
\(\sqrt{5} \times \sqrt{5}\) = \(\sqrt{5^2}\) = 5
What we notice here is that the square of -\(\sqrt{5}\) is 5 and square of \(\sqrt{5}\) is also 5. So we can say that the square roots of 5 are both \(\sqrt{5}\) and -\(\sqrt{5}\).
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 square roots of y.
Perfect Squares and Imperfect Squares
The square root of a perfect square number is always an integer. In contrast, the value of the square root of imperfect squares is a non-integer, that is, it contains decimals or fractions.
For example, 13 is the square root of 169, which is a perfect square, and 3.6055 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 higher imperfect square numbers.
Solved Square Root of 5 Examples
Example 1: Find the square root of 5 by the approximation of the prime number square roots.
Solution: Find the prime factorization of the number 5.
5 = 5 \(\times\) 1
\(\sqrt{5}\) = \(\sqrt{5 \times 1}\) [Apply square root both side]
⇒ \(\sqrt{5} \times \sqrt{1}\) [Use property \(\sqrt{ab}\) = \(\sqrt{a}.\sqrt{b}\)]
2.236 \(\times\) 1 [Substitute 1 for \(\sqrt{1}\) and 2.236 for \(\sqrt{5}\)]
⇒ 2.236 [Multiply]
So, the value of the square root of 5 is 2.236.
Example 2: Find the square root of 45.
Solution:
Use factor tree method to find the prime factors of 45
45 = 3 \(\times\) 3 \(\times\) 5
\(\sqrt{45}\) = \(\sqrt{3 \times 3\times 5}\)
\(\sqrt{45}\) = \(\sqrt{3^2 \times 5}\)
\(\sqrt{45}\) = 3\(\times \sqrt{5}\)
\(\sqrt{45}\) = 3 \(\times\) 2.236 = 6.708
So, the value of the square root of 45 is 6.708.
Example 3 : Evaluate the expression, 20 \(\sqrt{20}\) + 8.
Solution:
20 \(\sqrt{20}\) + 8
= 20 \(\times\sqrt{2 \times 2 \times 5}\) + 8
= 20 \(\times 2 \times \sqrt{5}\) + 8
= 40 \(\times\) 2.236 + 8 [ \(\sqrt{5}\) ≈ 2.236 ]
= 89.44 + 8 = 97.44
No, we can not find the square root of a negative number. This is because the square root of a negative number is what is known as an imaginary number.
A number is said to be a perfect square if the square root of the number is an integer.
Examples:
- 144 is a perfect square because \(\sqrt{144}\) = 12
- 81 is a perfect square because \(\sqrt{81}\) = 9
Yes , the product of two perfect squares is always a perfect square.
Example: Let’s consider two perfect square numbers, 36 and 25. The product of 36 and 25 is 900, which is a perfect square, and the square root of 900 is 30.
The measure of length is always positive. So the negative square root of 25 is neglected. Hence, we use just 5 as the measure of the side of a square field.