Question

To simplify: The given redical: An expression: displaystyle{sqrt[{{4}}]{{{left({x}^{2}-{4}right)}^{4}}}}. Use absolute value bars when necessary.

Rational exponents and radicals
ANSWERED
asked 2020-11-08
To simplify:
The given redical:
An expression: \(\displaystyle{\sqrt[{{4}}]{{{\left({x}^{2}-{4}\right)}^{4}}}}.\)
Use absolute value bars when necessary.

Expert Answers (1)

2020-11-09
Calculation:
Let simplify given radicals:
\(\displaystyle{\sqrt[{{4}}]{{{\left({x}^{2}-{4}\right)}^{4}}}}\)
\(\displaystyle\Rightarrow{\left({x}^{2}-{4}\right)}^{{\frac{4}{{4}}}}\ \text{(Applying rule} \displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}={a}^{{\frac{m}{{n}}}})\)
\(\displaystyle\Rightarrow{\left({x}^{2}-{4}\right)}^{1}\)
\(\displaystyle\Rightarrow{x}^{2}-{4}\)
Since a negative number raised to even power is always positive, so the solution of the given expression would be positive for any value of x.
So, we need to take absolute value of \(x^{2}\ -\ 4,\) to make the solution thue for all values of x.
Therefore, the simplified form of the given expression would be \(\displaystyle{\left|{x}^{2}-{4}\right|}.\)
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