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

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

2020-11-09
Calculation:
$$\displaystyle{\sqrt[{{4}}]{{{\left({x}^{2}-{4}\right)}^{4}}}}$$
$$\displaystyle\Rightarrow{\left({x}^{2}-{4}\right)}^{{\frac{4}{{4}}}}\ \text{(Applying rule} \displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}={a}^{{\frac{m}{{n}}}})$$
$$\displaystyle\Rightarrow{\left({x}^{2}-{4}\right)}^{1}$$
$$\displaystyle\Rightarrow{x}^{2}-{4}$$
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 $$\displaystyle{\left|{x}^{2}-{4}\right|}.$$

### Relevant Questions

Simplify each expression
(a) $$\displaystyle{\sqrt[{{4}}]{{{3}^{2}}}}$$
(b) $$\displaystyle{\sqrt[{{6}}]{{{\left({x}+{1}\right)}^{4}}}}$$
$$\displaystyle{\left({a}\right)}{\sqrt[{{6}}]{{{x}^{5}}}}$$
To simplify:
The expression $$\displaystyle{\sqrt[{{6}}]{{{x}^{5}}}}$$ and express the answer using rational exponents.
(b) $$\displaystyle{\left(\sqrt{{x}}\right)}^{9}$$
To simplify:
The expression $$\displaystyle{\left(\sqrt{{x}}\right)}^{9}$$ and express the answer using rational exponents.
To multiply:
The given expression. Then simplify if possible. Assume that all variables represent positive real numbers.
Given:
An expression: $$\displaystyle\sqrt{{3}}{\left(\sqrt{{27}}-\sqrt{{3}}\right)}$$
Use a right triangle to write the following expression as an algebraic expression. Assume that x is positive and in the domain of the given inverse trigonometric function.
Given:
$$\displaystyle \tan{{\left({{\cos}^{ -{{1}}}{5}}{x}\right)}}=?$$
Express as a trigonometric function of one angle.
a) $$\cos2\sin(-9)-\cos9\sin2$$
Find the exact value of the expression.
b) $$\sin\left(\arcsin\frac{\sqrt{3}}{2}+\arccos0\right)$$
a) Find the rational zeros and then the other zeros of the polynomial function $$\displaystyle{\left({x}\right)}={x}^{3}-{4}{x}^{2}+{2}{x}+{4},\ \tet{that is, solve}\ \displaystyle f{{\left({x}\right)}}={0}.$$
b) Factor $$f(x)$$ into linear factors.
$$\displaystyle{\sqrt[{{4}}]{{{c}{d}^{2}}}}\times{\sqrt[{{3}}]{{{c}^{2}{d}}}}.$$
$$\displaystyle{\sqrt[{{7}}]{{11}}}\times{\sqrt[{{6}}]{{13}}}$$
Use rational exponents to write a single radical expression $$\left(\sqrt[3]{x^{2}y^{5}}\right)^{12}$$
For the following exercises, use the compound interest formula, $$\displaystyle{A}{\left({t}\right)}={P}{\left({1}+\frac{r}{{n}}\right)}^{{{n}{t}}}$$.