Question

How do you simplify \[\sqrt[6]{256}\]?

Answer 1

Hint: The given question is about the simplification of the term. To simplify the term we need to divide the term \[256\] into multiple terms, we can divide the term by using the method prime factorization. The square root of a number is the multiplication of the same number. Here in the question, we need to find the sixth root of the term \ [256\]. The sixth root means multiplying the same number six times to get the term \[256\].

Complete step by step answer:
The given term is \[\sqrt[6]{256}\]
Any number can be divided by the basic number \[2\].
So, now let’s divide the term \[\sqrt[6]{256}\] into multiple terms \[2\].
As we know that
\[\Rightarrow {{2}^{6}}=64\]
Now let’s do the prime factorization for the number $ 256 $ with \[2\].
 $ \begin{align}
  & 2\left| \!{\underline {\,
  256 \,}} \right. \\
 & 2\left| \!{\underline {\,
  128 \,}} \right. \\
 & 2\left| \!{\underline {\,
  64 \,}} \right. \\
 & 2\left| \!{\underline {\,
  32 \,}} \right. \\
 & 2\left| \!{\underline {\,
  16 \,}} \right. \\
 & 2\left| \!{\underline {\,
  8 \,}} \right. \\
 & 2\left| \!{\underline {\,
  4 \,}} \right. \\
 & 2\left| \!{\underline {\,
  2 \,}} \right. \\
 & \,\,\,\left| \!{\underline {\,
  1 \,}} \right. _{{}}^{{}} \\
\end{align} $
So the number \[2\] is calculated $ 8 $ times to get the number $ 256 $ .
 $ \Rightarrow 256=2\times 2\times 2\times 2\times 2\times 2\times 2\times 2 $
We need the simplified term which can be multiplied six times to get the number $ 256 $ .
Now we can simplify the term \[\sqrt[6]{256}\]
\[\Rightarrow \sqrt[6]{256}=\sqrt[6]{{{2}^{6}}}\sqrt[6]{4}\]
As the simplified is in terms of \[2\] and $ 4 $.
Now again let’s simplify the above equation in the terms of \[2\].
\[\Rightarrow \sqrt[6]{256}=2.\sqrt[6]{{{2}^{2}}}\]
And we get the simplified term that is
\[\Rightarrow \sqrt[6]{256}=2.\sqrt[3]{2}\]
\[\therefore \sqrt[6]{256}=2.\sqrt[3]{2}\]
Hence the simplified term \[\sqrt[6]{256}\] is \[2.\sqrt[3]{2}\].

Note:
 Square root is inverse of the square. Squaring of numbers results in the higher value but the square root of several results from the lower value is known as simplified value. The square root is written as $ \dfrac{1}{2} $ in exponent form whereas the square is written as \[2\] In exponent form.
For example
Square of \[2\] means multiplying \[2\] two times.
\[\Rightarrow {{2} ^ {2}} =4\]
 $ \Rightarrow 2\times 2=4 $
The square root of \[4\] means dividing the number exactly into half.
\[\Rightarrow \sqrt{4}=2\]