Question

Simplify \[\sqrt[{10}]{{{x^8}\sqrt {{x^6}\sqrt {{x^{ - 4}}} } }}\] .

Answer 1

Hint: In order to solve the question given above, we have to use the properties of powers and bases. You need to keep a few things in mind while solving this question. Remember that whenever two same bases multiply their powers are added and when the same bases are in division then the powers are subtracted.

Complete step by step solution:
We are given that \[\sqrt[{10}]{{{x^8}\sqrt {{x^6}\sqrt {{x^{ - 4}}} } }}\] .
First, we have to solve the innermost problem.
As we can see that the power of \[x\] in the innermost under root is negative, therefore, we will reciprocal it to remove the negative sign.
On doing this we get,
\[\sqrt[{10}]{{{x^8}\sqrt {{x^6}\sqrt {\dfrac{1}{{{x^4}}}} } }}\].
Now,
\[\sqrt[{10}]{{{x^8}\sqrt {{x^6} \times \dfrac{1}{{{x^2}}}} }}\].
Since the bases are the same and they are in division the powers will be subtracted. We get,
\[
  \sqrt[{10}]{{{x^8}\sqrt {{x^{6 - 2}}} }} \\
   \Rightarrow \sqrt[{10}]{{{x^8}\sqrt {{x^4}} }} \\
 \] .
Now again, \[\sqrt[{10}]{{{x^8} \times {x^2}}}\].
We know that when the similar bases are in multiplication the powers are added.
On adding the powers, we get, \[\sqrt[{10}]{{{x^{10}}}}\] .
This can also be written as \[{\left( {{x^{10}}} \right)^{\dfrac{1}{{10}}}}\] .
Now, we get,
\[{x^{\dfrac{{10}}{{10}}}}\]
\[ \Rightarrow x\].
So, we get that, by simplifying are final answer is \[x\].

Note: While solving sums similar to the one given above, always remember the rules of powers with similar bases. To solve this question, we have merely subtracted and added the powers. The rule is that whenever two same bases multiply their powers are added and when the same bases are in division then the powers are subtracted. Also, remember when the power is in negative it can be reciprocated to remove the negative sign.