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Let's solve the expression step by step:

### Given Expression:
\[
\left( \frac{x^6 y}{y^2} \right)^{\frac{6}{3}}
\]

We are asked to find the exponents of \(x\) and \(y\) in the simplified expression \(\frac{x^r}{y^t}\).

### Step 1: Simplify the expression inside the parentheses
We start with the fraction inside the parentheses:
\[
\frac{x^6 y}{y^2}
\]
We can simplify the powers of \(y\):
\[
\frac{x^6 y}{y^2} = x^6 y^{1-2} = x^6 y^{-1}
\]

### Step 2: Apply the exponent \(\frac{6}{3}\)
Now, apply the exponent \(\frac{6}{3} = 2\) to the entire expression:
\[
\left( x^6 y^{-1} \right)^2 = x^{6 \cdot 2} y^{-1 \cdot 2} = x^{12} y^{-2}
\]

### Final expression:
\[
\frac{x^{12}}{y^2}
\]

### Step 3: Identify the exponents
- The exponent of \(x\), \(r = 12\).
- The exponent of \(y\), \(t = 2\).

So, the final answer is:
- \(r = 12\)
- \(t = 2\)

Do you want more details or have any further questions?

### Related Questions:
1. How do you simplify expressions with negative exponents?
2. What is the rule for multiplying exponents with the same base?
3. How do you handle exponents outside parentheses in a fraction?
4. What is the power of a power rule in exponents?
5. How would the result change if the exponent \(\frac{6}{3}\

Simplify the expression (x^6 * y / y^2)^(6/3) and find the exponents of x and y in the form x^r / y^t.

Learn how to simplify the expression (x^6 * y / y^2)^(6/3) using the laws of exponents. This problem covers key algebra concepts including power of a quotient, simplifying exponents, and applying multiplication rules for exponents. Suitable for students in Grades 8-10.

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