Let's analyze and simplify the given expression step-by-step. We are given the following expression:
[tex]\[
\sqrt[4]{x y^5} - \sqrt[4]{x^5 y}
\][/tex]
### Step 1: Understand the Expression
The expression involves two terms, each containing a fourth root. Let us denote these terms for clarity:
[tex]\[
\text{Term 1: } \sqrt[4]{x y^5}
\][/tex]
[tex]\[
\text{Term 2: } \sqrt[4]{x^5 y}
\][/tex]
### Step 2: Rewrite Each Term in Exponential Form
We can express roots in terms of fractional exponents:
[tex]\[
\sqrt[4]{x y^5} = (x y^5)^{\frac{1}{4}}
\][/tex]
[tex]\[
\sqrt[4]{x^5 y} = (x^5 y)^{\frac{1}{4}}
\][/tex]
### Step 3: Simplify Each Term
Next, simplify the expression within each exponent:
[tex]\[
(x y^5)^{\frac{1}{4}}
\][/tex]
[tex]\[
(x^5 y)^{\frac{1}{4}}
\][/tex]
### Step 4: Apply the Power of a Product Rule
The power of a product rule states that [tex]\((ab)^c = a^c b^c\)[/tex]. Applying this rule, we get:
[tex]\[
(x y^5)^{\frac{1}{4}} = x^{\frac{1}{4}} (y^5)^{\frac{1}{4}} = x^{\frac{1}{4}} y^{\frac{5}{4}}
\][/tex]
[tex]\[
(x^5 y)^{\frac{1}{4}} = (x^5)^{\frac{1}{4}} y^{\frac{1}{4}} = x^{\frac{5}{4}} y^{\frac{1}{4}}
\][/tex]
### Step 5: Combine the Terms into the Original Subtraction
Now that we have simplified each term, we can substitute them back into the original expression:
[tex]\[
(x^{\frac{1}{4}} y^{\frac{5}{4}}) - (x^{\frac{5}{4}} y^{\frac{1}{4}})
\][/tex]
### Step 6: Present the Final Simplified Form
Thus, the simplified form of the given expression is:
[tex]\[
\sqrt[4]{x y^5} - \sqrt[4]{x^5 y} = (x y^5)^{\frac{1}{4}} - (x^5 y)^{\frac{1}{4}}
\][/tex]
So, the final result is:
[tex]\[
(x y^5)^{\frac{1}{4}} - (x^5 y)^{\frac{1}{4}}
\][/tex]
