Square and Square Root
As described earlier, the square root of a number is a number that, when multiplied by itself, results in the original number. Hence, the process for finding the square root of a number is the opposite of finding the square, the main difference being whether we multiply or divide.
In general, if a number ‘X’ is multiplied by itself to get Y, the square of X is denoted by X\(^2\), which is equal to Y. On the other hand, the square root of Y is denoted by \(\sqrt{Y}\), which is equal to X.
Mathematically, we always represent squares and square roots as, X\(^2\) = Y and \(\sqrt{Y}\) = X.
Note : The square root of a number is denoted by \(\sqrt{~~}\) .
Square Root of 1024 using Prime Factorization
The following steps can be applied to determine the square root of any number using the prime factorization method. In this case, this method is applied to find \(\sqrt{1024}\).
Step 1: Find the prime factors of 1024
∴ 1024 = 2 × 2 × 2 × 2 × 2 × 2 × 2 x 2 × 2 x 2
Step 2: Make pairs of identical factors.
1024 = (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2)
Step 3: Take a factor from each pair and multiply them to get the square root.
2 x 2 x 2 x 2 x 2 = 32
So the square root of 1024 is 32.
The above steps can be summarized as:
\(\sqrt{1024}\) = \(\sqrt{(2~\times~2)~~\times~~(2~\times~2)~~\times~~(2~\times~2)~~\times~~(2~\times~2)~~\times~~(2~\times~2)}\)
\(\Rightarrow\) \(\sqrt{1024}\) = 2 × 2 × 2 × 2 × 2
\(\Rightarrow\) \(\sqrt{1024}\) = 32
Definition of Negative Square Root
Every positive number has two square roots, one is positive and other is negative as their absolute value is the same.
Let us understand this with example,
32 × 32 = 1024
-32 × -32 = 1024 [Since, (-) x (-) = (+)]
From the above the square of -32 is 1024 and square of 32 is also 1024, so we can say that square roots of 1024 are both 32 and -32.
\(\Rightarrow~~\sqrt{1024}\) = ± 32
In general,
Solved Square Root of 1024 Examples
Example 1: The area of the square shaped table top is 1024 m\(^2\). Find the length of the side of the table.
Solution:
Area of square shaped table = 1024
Let the side of table be ‘a‘
Since, the area of the square is given by (side)\(^2\),
Area of table top = a\(^2\)
\(\Rightarrow ~1024\) = \(a^2\) [Substitute the value of area]
\(\Rightarrow ~\sqrt{1024}=\sqrt{a^2} \) [Taking square root on both sides]
∴ a = 32 [Simplify]
Therefore, the length of the side of the table top is 32 m.
Example 2: Evaluate 5 \(\sqrt{1024}\) ÷ 8
Solution:
Given expression: 5 \(\sqrt{1024}\) ÷ 8
\(\frac{5\sqrt{1024}}{8}\) [Rewrite the expression]
= \(\frac{5~\times~32}{8}\) [Substitute \(\sqrt{1024}\) = 32]
= 5 × 4 [Simplify]
= 20 [Multiply]
Hence the value of given expression 5 \(\sqrt{1024}\) ÷ 8 is 20.
Example 3: Find the value of 6\(\sqrt{1024}\) – 3.
Solution:
6\(\sqrt{1024}\) – 3 [Write the expression]
= 6 x 32 -3 [Substitute \(\sqrt{1024}\) = 32]
= 192 – 3 [Multiply]
= 189 [Subtract]
So, 6\(\sqrt{1024}\) – 3 = 189
We know that once we use factorization we find that the square root of 1024 is either 32 or -32.
Since square root of 1024 is given by 1024 written under the radical symbol.
So in exponential form it is written as 1024 raised to the power of half or one by two.
Yes, 1024 is a perfect square; as the square root of 1024 is 32, and 32 is an integer.
The square root of 1024 in radical form is 1024 along with the radical sign.