To solve such problems we need to know about expression, power, and exponents formulas.
Expression
In mathematics, an expression is defined as a set of numbers and mathematical operations that is formed according to rules which is dependent on the context.
Power
Power is a number(a) that is raised to another number(b) which helps us to know how many numbers (a) of times the number(b) is been multiplied by itself. such as [tex]\bold{a^b}[/tex].
Few Exponents formulas
- [tex]a^b \times a^c = a^{b+c}[/tex]
- [tex]\dfrac{a^b}{a^c} = a^{b-c}[/tex]
- [tex]a^{\frac{a}{b}} = \sqrt[b]{x^a}[/tex]
- [tex](a^m)^n = a^{m\times n}[/tex]
Given to us,
- [tex]x^{\frac{1}{6}}\cdot x^{\frac{1}{6}} = \sqrt[3]{x}[/tex]
- [tex]x^{\frac{2}{3}} = \sqrt[3]{x^2}[/tex]
- [tex]x^{\frac{5}{6}} = \sqrt[5]{x^6}[/tex]
- [tex]\dfrac{x^{\frac{3}{4}}}{x^{\frac{1}{2}}} = \sqrt[4]{x}[/tex]
a.) [tex]x^{\frac{1}{6}}\cdot x^{\frac{1}{6}} = \sqrt[3]{x}[/tex]
[tex]x^{\frac{1}{6}}\cdot x^{\frac{1}{6}} = \sqrt[3]{x}\\\\x^{\frac{1}{6}+\frac{1}{6}} = \sqrt[3]{x}\\\\x^{\frac{2}{6}} = \sqrt[3]{x}\\\\x^{\frac{1}{3}} = \sqrt[3]{x^1}\\\\x^{\frac{1}{3}} = \sqrt[3]{x}[/tex]
This rational exponent expression is simplified correctly as a radical expression.
b.) [tex]x^{\frac{2}{3}} = \sqrt[3]{x^2}[/tex]
[tex]x^{\frac{2}{3}} = \sqrt[3]{x^2}\\\\[/tex]
This rational exponent expression is simplified correctly as a radical expression.
c.) [tex]x^{\frac{5}{6}} = \sqrt[5]{x^6}[/tex]
[tex]x^{\frac{5}{6}} = \sqrt[6]{x^5}\\\\[/tex]
[tex]x^{\frac{5}{6}} \neq \sqrt[5]{x^6}\\\\[/tex]
This rational exponent expression is not simplified correctly as a radical expression.
d.) [tex]\dfrac{x^{\frac{3}{4}}}{x^{\frac{1}{2}}} = \sqrt[4]{x}[/tex]
[tex]\dfrac{x^{\frac{3}{4}}}{x^{\frac{1}{2}}} = \sqrt[4]{x}\\\\x^{(\frac{3}{4}-\frac{1}{2})} = \sqrt[4]{x}\\\\x^{(\frac{3-2}{4})} = \sqrt[4]{x}\\\\x^{(\frac{1}{4})} = \sqrt[4]{x}[/tex]
This rational exponent expression is simplified correctly as a radical expression.
Learn more about Exponents formulas:
https://brainly.com/question/24346604
