If 3^{x} = 81 and 5^{x+y} = 625, then the value of x and y are :

Question

Accepted Answer

To solve the problem and justify why Option 4 is correct, we need to evaluate the given equations:

1. **Equation 1:** $3^x = 81$
2. **Equation 2:** $5^{x+y} = 625$

**Step-by-step Explanation:**

1. **Solving Equation 1:**

- We know that $81$ can be expressed as a power of $3$. Specifically, $81 = 3^4$.
   - Therefore, $3^x = 3^4$.
   - By comparing the exponents, we find that $x = 4$.

2. **Solving Equation 2:**

- We know that $625$ can be expressed as a power of $5$. Specifically, $625 = 5^4$.
   - Therefore, $5^{x+y} = 5^4$.
   - Since we have already determined that $x = 4$, we substitute this value into the equation: $5^{4+y} = 5^4$.
   - By comparing the exponents, we find that $4 + y = 4$.
   - Solving for $y$, we get $y = 0$.

Thus, the values of $x$ and $y$ are $x = 4$ and $y = 0$, which corresponds to Option 4.

**Explanation of Why Other Options are Incorrect:**

1. **Option 1: $x = 5, y = 1$**
   - If $x = 5$, then $3^x = 3^5$, which is not equal to $81$. Therefore, this option is incorrect.

2. **Option 2: $x = 2, y = 0$**
   - If $x = 2$, then $3^x = 3^2 = 9$, which is not equal to $81$. Therefore, this option is incorrect.

3. **Option 3: $x = 3, y = 1$**
   - If $x = 3$, then $3^x = 3^3 = 27$, which is not equal to $81$. Therefore, this option is incorrect.

By following the logical steps and calculations, we confirm that Option 4 is indeed the correct answer, with $x = 4$ and $y = 0$.