What is the Square Root of 8?
Rounding to eight decimal places, the square root of 8 is 2.82842712.
The square root of 8 results in a number that when multiplied by itself gives the value 8. Also, the square root of 8 can be written in different ways:
Square root of 8 in radical form: \(\sqrt{8}\) or \(\sqrt{2}\)
Square Root of 8 in decimal form: 2.828
Square root of 8 in exponent form: (8)\(^{\frac{1}{2}}\) or (8)\(^{0.5}\)
How do we find the Square Root of 8?
We can find the square root of a number by using the prime factorization method. Let us apply this method to find \(\)\sqrt{8}[/latex .
Step 1: Find the prime factors of 8
Prime factorization of 8 = 2 × 2 × 2.
Step 2: Make pairs of identical factors.
Write the third 2 as \(\sqrt{2}\) x \(\sqrt{2}\)
8 = (2 × 2) × (\(\sqrt{2}\) x \(\sqrt{2}\))
Step 3: Take a factor from each pair and multiply them to get the square root.
\(\sqrt{8}\) = 2 × \(\sqrt{2}\)
\(\Rightarrow\) \(\sqrt{8}\) = 2\(\sqrt{2}\)
Since the value of \(\sqrt{2}\) = 1.414 (approx.)
\(\Rightarrow\) \(\sqrt{8}\) = 2 × 1.414 = 2.828 (approx.)
So, the square root of 8 is 2\(\sqrt{2}\) or about 2.828.
Note: The simplest radical form of square root of 8 is 2\(\sqrt{2}\).
Definition of Negative Square Root
Each positive number has two square roots, one is positive and the other is negative, as both the positive and negative numbers have the same absolute value.
Let us understand this with an example,
2\(\sqrt{2}\) x 2 \(\sqrt{2}\) = 8
(-2\(\sqrt{2}\) ) × (-2\(\sqrt{2}\)) = 8
[Since, multiplying two negatives results in a positive]
From the above the square of –2\(\sqrt{2}\) is 8 and the square of 2\(\sqrt{2}\) is also 8. So, we can say that square roots of 8 are both 2\(\sqrt{2}\) and –2\(\sqrt{2}\).
Hence, \(\sqrt{8}\) = ± 2\(\sqrt{2}\)
Is the Square Root of 8 Rational or Irrational?
A rational number is one that can be written as \(\frac{p}{q}\), where p and q are integers and q is not equal to 0.
Irrational numbers are those that are not rational, that is, they cannot be expressed as a fraction. Irrational numbers are non-terminating and non-repeating values.
Let’s now consider the square root of 8. The decimal equivalent of \(\sqrt{8}\) is 2.828427124….
The number 2.828427124… is a non-terminating decimal with non-repeating digits. Thus, the number 2.828427124… cannot be written in the \(\frac{p}{q}\) format.
Hence, the square root of 8 is an irrational number.
Rapid Recall
Solved Square Root of 8 Examples
Example 1: Lily wishes to decorate the square shaped wall of her room. The wall is spread over 8 square feet. How long is the wall’s length on each side?
Solution:
Given, the area of the wall is 8 square feet.
We know that,
The area of a square = side x side
So, 8 = side\(^2\)
To find the side of the square wall, we will have to find the square root of 8, that is,
\(\sqrt{8}\) = ± 2\(\sqrt{2}\).
We already know that the length cannot be negative.
Hence, the side length of the wall is 2\(\sqrt{2}\) feet.
Example 2: Find the value of 5 multiplied by \(\sqrt{8}\).
Solution:
5 × \(\sqrt{8}\) = ?
Substitute 2.828 for \(\sqrt{8}\)
5 × \(\sqrt{8}\) = 5 × 2.828
= 14.14 [Multiply]
Hence, the value of 5 multiplied by \(\sqrt{8}\) is about 14.14.
Example 3: If the square Root of 8 is 2.828, find the value of the square root of 0.08.
Solution:
Let’s write \(\sqrt{0.08}\) in \(\frac{p}{q}\) form.
That is, \(\sqrt{\frac{8}{100}}\)
= \(\frac{\sqrt{8}}{\sqrt{100}}\)
= \(\frac{2.828}{10}\) [Square root of 100 is 10]
= 0.2828 [Simplify]
Hence, the value \(\sqrt{0.08}\) is 0.2828
The square root of 8 is not an integer. So this indicates that 8 is not a perfect square.
We can find the square root of 8 in two ways. They are:
- Prime factorization method.
- Long division method for both perfect and non-perfect squares.
The square root of 8 is about 2.82842.
8 has two square roots just like any other number.