Question

How do you simplify \[\sqrt[4]{162}\] ?

Answer 1

Hint: In the above question we have to simplify the given radical expression. There are two ways to simplify the radical expression. In this case we will use the identity \[\sqrt[4]{{{a}^{4}}}=a\] . First, we have to find the prime factors of 162 using the prime factorization method. After that we have to apply simple exponential rules to get the answer.

Complete step by step answer:
The above question belongs to the concept of simplifying radical expression. Radical expression is an expression which contains root, it can be square root, cube root of nth root. A radical is basically a root of the equation. To solve this question, we will use the prime factorisation method. Firstly, prime numbers are the numbers which have only two multiples, 1 and the number itself. In the prime factorization method we will write numbers in the form of its prime factors.
Now in the given question we have to simplify it. First, we will reduce 162 into its prime factors. Then we will simplify it further in order to reduce it into identity \[\sqrt[4]{{{a}^{4}}}=a\] .
Here we can write 162 as,
\[\begin{align}
  & 162=3\times 3\times 3\times 3\times 2 \\
 & 162={{3}^{4}}\times 2 \\
\end{align}\]
\[\begin{align}
  & \sqrt[4]{162}=3\times 3\times 3\times 3\times 2 \\
 & \Rightarrow \sqrt[4]{162}={{3}^{4}}\times 2 \\
 & \Rightarrow 162={{({{3}^{4}}\times 2)}^{\dfrac{1}{4}}} \\
 & \Rightarrow 162={{({{3}^{4}})}^{\dfrac{1}{4}}}\times {{(2)}^{\dfrac{1}{4}}} \\
 & \Rightarrow 162=3\times {{(2)}^{\dfrac{1}{4}}} \\
 & \Rightarrow 162=3\sqrt[4]{2} \\
\end{align}\]
Therefore, on simplification we get \[3\sqrt[4]{2}\] .

Note:
While solving the expression make sure that the argument or the value inside the root is positive because that value will be imaginary which doesn’t make any sense in case of radical expression. Carefully find the prime factors of the given inside the root. Perform calculations carefully in order to avoid mistakes.