Sure, let's solve the given expression step-by-step.
We are given the expression in the form:
[tex]\[ V(5) = \sqrt[5]{5x^6 - 7x^5 + 3x^4} \][/tex]
The objective is to find [tex]\( V(5) \)[/tex], which represents the fifth root of the polynomial [tex]\( 5x^6 - 7x^5 + 3x^4 \)[/tex].
### Step 1: Identify the polynomial inside the fifth root
First, we identify the polynomial expression inside the fifth root:
[tex]\[ P(x) = 5x^6 - 7x^5 + 3x^4 \][/tex]
### Step 2: Understand the fifth root operation
Next, we interpret the fifth root of the polynomial. Taking the fifth root can be represented mathematically as raising the expression to the power of [tex]\( \frac{1}{5} \)[/tex]:
[tex]\[ V(5) = \left( 5x^6 - 7x^5 + 3x^4 \right)^{1/5} \][/tex]
### Step 3: Present the simplified form of the solution
The simplified form of the expression is:
[tex]\[ V(5) = \left(5x^6 - 7x^5 + 3x^4\right)^{1/5} \][/tex]
This effectively means that we take the polynomial [tex]\( P(x) = 5x^6 - 7x^5 + 3x^4 \)[/tex], and find its fifth root.
### Final Solution
Thus, the solution for [tex]\( V(5) \)[/tex] is:
[tex]\[ V(5) = \sqrt[5]{5x^6 - 7x^5 + 3x^4} \][/tex]
Or more neatly written as:
[tex]\[ V(5) = (5x^6 - 7x^5 + 3x^4)^{1/5} \][/tex]
