What is Square Root of 32?
The square root of a number is a value that, when multiplied by itself, gives us the original number, as introduced earlier. Hence, finding the square root is the converse of finding the square of a number.
The square root of 32 is written as \(\sqrt{32},\) with the radical sign ‘\(\sqrt{~~}\)’ and the radicand being 32. The square root of 32 has a value that is nearly equal to 5.6568542494… and this value is non-terminating and non-repeating.
How did we obtain the value of \(\sqrt{32}\) as 5.6568542494…? It can be obtained using a couple of methods, but to find the value of the number with precision, the long division method is used. For now, we will use the prime factorization method.
The number \(\sqrt{32}\) can be expressed as 4\(\sqrt2\) in its simplest form. We will use prime factorization to obtain the result.
Finding the Square Root of 32
Step 1: Prime factorization of 32 = \(2\times 2\times 2\times 2\times 2\)
Step 2: Make pairs of identical factors.
Since we do not have a pair of last 2 we can write 2 as \(\sqrt{2}\times \sqrt2\)
32 = \(2\times2\times\sqrt2\times\sqrt2\)
Step 3: Take a factor from each pair and multiply them to get the square root.
\(\sqrt{32}\ = 4\times\sqrt2\)
Since the value of \(\sqrt2\) = 1.4142135623 (approx.)
So \(\sqrt{32}\) = 4 \(\times\) 1.4142135623 = 5.6568542492(approx.)
So, the square root of 32 is 4\(\sqrt2\) or about 5.6568542492
Positive and Negative Square Root of a Number
The square root of a number will be positive or negative, having the same absolute value.
Let us understand this in context with the square root of 32.
4\(\sqrt2\times4 \sqrt2\) = 32
\((-4\sqrt2)\times\ (-4\sqrt2)\) = 32
[Since, the product of two negative integers results in a positive integer.]
From this we can understand that the square of -4\(\sqrt2\) is 32 and square of 4\(\sqrt2\) is also 32, so we can say that square roots of 32 are both 4\(\sqrt2\) and \(-4\sqrt2.\)
Hence, \(\sqrt{32}=\pm\ 4\sqrt2\)
This is the simplest form of presenting the value of \(\sqrt{32}\) .
In general,
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, 2 is the square root of 4, which is a perfect square, and 1.414 is the square root of 2, which is an imperfect square.
Note: The table lists the approximate value of the square roots of a few numbers that can be memorized to determine the square roots of larger imperfect square numbers.
Number | Square root |
|---|---|
2 | 1.414 |
3 | 1.732 |
5 | 2.236 |
6 | 2.449 |
7 | 2.646 |
8 | 2.828 |
10 | 3.162 |
Using this table, we can find a more precise result for \(\sqrt{32}\):
\(\sqrt{32}\) = \(4\sqrt{2}\) = 4 \(\times\) 1.414213 = 5.65685
Rapid Recall
Solved Square Root of 32 Examples
Example 1: Solve for \(x: x^2-32=0\)
Solution:
\(x^2-32=0\)
\(x^2=32\) [Add 32 on both sides]
\(x=\sqrt{32}\) [Take the square root on both sides]
\(x=\pm4\sqrt2\)
Therefore, \(x=\pm4\sqrt2\).
Example 2: Simplify the expression \(4\sqrt2-\sqrt{1016-(520+464)}\)
Solution:
\(=4\sqrt2-\sqrt{1016-(520+464)}\)
\(=4\sqrt2-\sqrt{1016-(984)}\) [Add]
\(=4\sqrt2-\sqrt{32}\) [Subtract]
\(=4\sqrt2-4\sqrt2=0\) [Write \(\sqrt{32}\) as \(4\sqrt2\)]
Therefore \(4\sqrt2-\sqrt{1016-(520+464)}=0\)
Example 3:
The cost ‘\(x\)’ of making a square shaped window with a side length of n inches can be represented in the form of the equation:
\(x={\frac{n^2}{8}+261}\)A window costs $265. What is the side length of the window?
Solution:
We have been given the cost of the window in terms of its side length \(n\).
We are also provided with the cost of making a square shaped window. We can equate the cost of making the window to the equation.
265 = \({\frac{n^2}{8}+261}\)
265 – 261 = \({\frac{n^2}{8}}\) [Subtract 261 from both sides]
4 = \({\frac{n^2}{8}}\)
32 = \(n^2\) [Multiply both sides by 8]
\(n=\pm\sqrt{32}\) [Take square root on both sides]
The length of a side cannot be negative, so the length of the side of the window is \(\sqrt{32}\) inches or about 5.65 inches.
Example 4 : Evaluate each expression
(a) \(9\sqrt{32}+10\)
(b) \(20\sqrt{69}+8\)
Solution:
(a) \(9\sqrt{32}+10\)
= \(9\times5.65+10\) [ Value of \(\sqrt{32}\) = 5.65]
= 50.85 + 10 = 60.85
Hence, the value of \(9\sqrt{32}+10\) is 60.85.
(b) \(20\sqrt{69}+8\)
= \(20\times\sqrt{3\times32}+8\)
= \(20\times\sqrt3\times\sqrt{32}+8\)
= \(20\times1.732\times5.65+8\) [Value of \(\sqrt3\approx1.732\) and \(\sqrt{32}=5.65\)]
= 195.716 + 8
= 203.716
Hence, the value of \(20\sqrt{69}+8\) is 203.716.
Example 5: The area of a circular crop field is 32𝝿 ft2. Find the radius of the field.
Solution :
Let the radius of the circular crop field be ‘r’ ft.
The area of a field = 𝝿 r\(^2\)
32𝝿 = 𝝿 r\(^2\) [area = 32𝝿 ft\(^2\)]
32 = r\(^2\) [Divide both sides by 𝝿]
32 = r
5.65 = r
Hence, the radius of the circular field is 5.65 ft.
The square root of 32 is expressed as 32 raised to the power of half in the exponential form.
The long division method provides the most accurate result of a square root of a number.
A symbol that is a combination of the plus and minus signs, shows us that there are two results, where one is positive and the other is negative with the same absolute value.
The square root of a number is represented by placing the number under the radical sign.
No, we can not find the square root of a negative number.