To identify the rational exponent expression for the fourth root of [tex]\( f \)[/tex], let’s start by understanding the concept of roots and exponents in mathematics.
When we talk about the [tex]\( n \)[/tex]-th root of a number [tex]\( x \)[/tex], it means finding a number that, when raised to the power [tex]\( n \)[/tex], equals [tex]\( x \)[/tex]. Mathematically, the [tex]\( n \)[/tex]-th root of [tex]\( x \)[/tex] is represented as:
[tex]\[ \sqrt[n]{x} = x^{\frac{1}{n}} \][/tex]
In this specific problem, we are dealing with the fourth root of [tex]\( f \)[/tex]. This is represented as:
[tex]\[ \sqrt[4]{f} \][/tex]
Using the rule for roots and exponents, we can rewrite the fourth root of [tex]\( f \)[/tex] in its rational exponent form:
[tex]\[ \sqrt[4]{f} = f^{\frac{1}{4}} \][/tex]
Now let's evaluate the given options to determine which one matches the rational exponent form of the fourth root of [tex]\( f \)[/tex]:
1. [tex]\( f^{\frac{1}{4}} \)[/tex]
2. [tex]\( f^4 \)[/tex]
3. [tex]\( 4f \)[/tex]
4. [tex]\( \frac{f}{4} \)[/tex]
The correct exponent form is [tex]\( f^{\frac{1}{4}} \)[/tex]. This directly represents the fourth root of [tex]\( f \)[/tex], as shown above.
Thus, the rational exponent expression of [tex]\( \sqrt[4]{ f } \)[/tex] is:
[tex]\[ f^{\frac{1}{4}} \][/tex]
