Square root:
The square root of a number is a value that, when multiplied by itself, gives the original number.
In general, if \(y=x^2\), then \(x\) is the square root of \(y\). The square root of a number is denoted by \(\sqrt{}\)
Important!
Every perfect square has two integer square roots. One is positive and the other is negative. In general, \(\sqrt{n^2} =±n\). In this chapter, we will consider the positive square root alone.
For example: \(8\times 8 = 64 \) and \((-8)\times (-8) = 64\). So the \(\sqrt{64} = \pm 8\)
Square root by the repeated subtraction method:
The square root of a perfect square number can be found by successively subtracting consecutive odd numbers starting from \(1\) until you reach \(0\) as an answer.
The total number of subtraction steps required to reach exactly \(0\) is equal to the square root of that number.
Example:
\(16 - 1=15\), \(15-3=12\), \(12-5=7\), \(7-7=0\).
Here \(4\) steps are there to attain \(0\).
Therefore, \(\sqrt{16} = 4\)
Steps to find the square root of a number:
Step 1: Write the given natural number as a product of prime factors.
Step 2: Group the factors in pairs so that both factors in each pair are equal.
Step 3: Now, see whether some factors are left over or not. If no factor is left over in grouping, then the given number is a perfect square. Otherwise, it is not a perfect square.
Step 4: Take one factor from each group and multiply them to obtain the number whose square is the given number.
Let's see an example to understand this concept clearly.
Example:
Find \(\sqrt{324}\).
We need to find the value of \(\sqrt{324}\).
Step 1: Write \(324\) as a product of prime factors.
\(324 = 2 \times 2 \times 3 \times 3 \times 3 \times 3\)
Step 2: Group the prime factors.
\(324 = (2 \times 2) \times (3 \times 3) \times (3 \times 3)\)
Step 3: Here, no factor is left over in the grouping.
So, the given number is a perfect square.
Step 4: Now, take one factor common to each group.
\(\sqrt{324}\) \(=\) \(2 \times 3 \times 3\)
\(\sqrt{324}\) \(=\) \(18\)
Therefore, the square root of \(\sqrt{324}\) is \(18\).
Closest Square Numbers:
A perfect square has a whole number as its square root. When a number is not a perfect square, its square root can be estimated using a nearby perfect square.
Step to find the closest square number:
Step 1: Identify the two consecutive perfect squares between which the given number lies.
Step 2: Find their square roots.
Step 3: The square root of the given number lies between these two whole numbers.
Step 4: Determine which perfect square is closer to the given number to get a better estimate.
Let's see an example to understand this concept clearly.
Example:
Find the closest square value of \(250\).
Step 1: We know that \(100 < 250 < 400\) and \(\sqrt{100} = 10\) and \(\sqrt{400} = 20\).
So, \(10 < \sqrt{250} < 20\).
But we are still not very close to the number whose square is \(250\).
Step 2: To find the square root of the number.
We know that \(15^2 = 225\) and \(16^2 = 256\)
Step 3: The square root of the given number lies between these two whole numbers.
Therefore, \(15 < \sqrt{250} < 16\).
Step 4: To determine the closest square number.
Since \(256\) is much closer to \(250\) than \(225\).
Therefore, \(\sqrt{250}\) is approximately equal to \(16\).
Reference:
National Council of Educational Research and Training (2025). Maths - Standard 8. Ganita Prakash, Part I. A square and a cube (pg.7 - 10). Published at the Publication Division by the Secretary, National Council of Educational Research and Training, Sri Aurobindo Marg, New Delhi.