Question

How do you simplify radical $8$?

Answer 1

Hint: Radicals also called "roots" are the "opposite" operation of applying exponents; we can reverse a power with a radical, and we can reverse a radical with a power. For instance, if we square $2$ , we get $4$ , and if we " take the square root of $4$ " , we get $2$ ; if we square $3$ , we get $9$ , and if we " take the square root of $9$ " , we get $3$ . In mathematical notation, the previous sentence means the following:
$\begin{align}
  & {{2}^{2}}=4,so\sqrt{4}=2 \\
 & {{3}^{2}}=9,so\sqrt{9}=3 \\
\end{align}$
The " $\sqrt{...}$ " symbol used above is called the " radical " symbol . The expression " $\sqrt{8}$ " is read as " root eight ", " radical eight ", or " the square root of eight ".
To simplify a term containing a square root, we " bring out " anything that is a "perfect square", that is, we factor the number inside the radical symbol and then we take out anything in front of that symbol that has two copies of the same factor. For example, is the square of $3$, so the square root of $9$ contains two copies of the factor $3$; thus, we can take one $3$ out front, leaving nothing except $1$ inside the radical symbol.

Complete step by step solution:
The number whose radical is to be found is $8$
The radical form of the number $8$ is $\sqrt{8}$
Now we have to simplify this.
To do this, first, find the prime factorization of the number
Prime factorization of $8$ is $2\times 2\times 2$
Therefore, $\sqrt{8}=\sqrt{2\times 2\times 2}$
From here we can see that 8 is not a perfect square of a number but it is a perfect cube.
Now for finding the square root of 8, we pair two 2's and take out a $2$ with one $2$ left inside the radical symbol.
Thus , we get
$\sqrt{8}=2\sqrt{2}$ which is not a whole number
While $8={{2}^{3}}$
Thus $\sqrt[3]{8}=2$ which is a whole number
Therefore , the radical of $8$ is $2\sqrt{2}$ since $2\sqrt{2}$ is a square root of 8 .
This implies $\sqrt{8}=2\sqrt{2}$ and the value of $\sqrt{2}=1.414...$

Hence the radical of $8$ is $2\times 1.414..=2.828$.

Note: In mathematics, the n-th root of a number $x$ is a number $r$ which when raised to the power $n$ gives $x$ where $n$ is a positive integer, sometimes called the degree of a root.An unresolved root especially the one using the radical symbol is sometimes referred to as a surd or a radical.