Express the radical as power. 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.

Question
$$\displaystyle{\left({a}\right)}{\sqrt[{{6}}]{{{x}^{5}}}}$$
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.

2020-11-06
(a) Concept used:
If ais a real number, n is a positive integer and $$root(n)(a^m)\ \text{is a real number then the rational exponent expression}\ \displaystyle{\sqrt[{{n}}]{{{a}^{m}}}}\ \text{is equivalent to radical expression}\ \displaystyle{a}^{{\frac{m}{{n}}}}.$$
The above statement can be expressed as,
$$\displaystyle{\sqrt[{{n}}]{{a}}}^{m}={a}^{{\frac{m}{{n}}}}$$ ...... (1)
Calculation:
The given expression is $$\displaystyle{\sqrt[{{6}}]{{{x}^{5}}}}$$
Subtitute 6 for n, 5 for m and x for a in the equation (1) to obtain the equialent expression in rational exponent as,
$$\displaystyle{\sqrt[{{6}}]{{{x}^{5}}}}={\left({x}^{{\frac{1}{{6}}}}\right)}^{5}$$
$$\displaystyle={x}^{{\frac{5}{{6}}}}$$
Conclusion:
Thus, the equvalent expression of the expression $$\displaystyle{\sqrt[{{6}}]{{{x}^{5}}}}{i}{s}{x}^{{\frac{5}{{6}}}}.$$
(b) Calculation:
The given expression is $$\displaystyle{\left(\sqrt{{x}}\right)}^{9}$$
Subtitute 2 for n, 9 for m and x for a in the equation (1) to obtain the equialent expression in rational exponent as,
$$\displaystyle{\left(\sqrt{{x}}\right)}^{9}={\left({x}^{{\frac{1}{{2}}}}\right)}^{9}$$
$$\displaystyle={x}^{{\frac{9}{{2}}}}$$
Conclusion:
Thus, the equivalent expression of the expression $$\displaystyle{\left(\sqrt{{x}}\right)}^{9}{i}{s}{x}^{{\frac{9}{{2}}}}.$$

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