If $ ext{If } \left(\frac{5}{7} ight)^4 imes \left(\frac{5}{7} ight)^{-3} = \left(\frac{5}{7} ight)^{5x-2}$, then x is-

Q: If $\text{If } \left(\frac{5}{7}\right)^4 \times \left(\frac{5}{7}\right)^{-3} = \left(\frac{5}{7}\right)^{5x-2}$, then x is-

To solve the given equation $\left(\frac{5}{7}\right)^4 \times \left(\frac{5}{7}\right)^{-3} = \left(\frac{5}{7}\right)^{5x-2}$, we need to simplify the left-hand side and equate the exponents. ### Step-by-Step Explanation: 1. **Simplify the Left-Hand Side:** - The expression $\left(\frac{5}{7}\right)^4 \times \left(\frac{5}{7}\right)^{-3}$ can be simplified using the property of exponents: $a^m \times a^n = a^{m+n}$. - Applying this property, we get: \[ \left(\frac{5}{7}\right)^4 \times \left(\frac{5}{7}\right)^{-3} = \left(\frac{5}{7}\right)^{4 + (-3)} = \left(\frac{5}{7}\right)^1 \] 2. **Equate the Exponents:** - Now, we have $\left(\frac{5}{7}\right)^1 = \left(\frac{5}{7}\right)^{5x-2}$. - Since the bases are the same, we can equate the exponents: \[ 1 = 5x - 2 \] 3. **Solve for $x$:** - Add 2 to both sides of the equation: \[ 1 + 2 = 5x \] \[ 3 = 5x \] - Divide both sides by 5 to solve for $x$: \[ x = \frac{3}{5} \] ### Conclusion: - The value of $x$ is $\frac{3}{5}$, which corresponds to Option 3. ### Explanation of Incorrect Options: 1. **Option 1: $\frac{1}{5}$** - If $x = \frac{1}{5}$, then $5x = 1$, and $5x - 2 = 1 - 2 = -1$, which does not satisfy the equation $1 = 5x - 2$. 2. **Option 2: $\frac{2}{5}$** - If $x = \frac{2}{5}$, then $5x = 2$, and $5x - 2 = 2 - 2 = 0$, which does not satisfy the equation $1 = 5x - 2$. 3. **Option 4: $\frac{4}{5}$** - If $x = \frac{4}{5}$, then $5x = 4$, and $5x - 2 = 4 - 2 = 2$, which does not satisfy the equation $1 = 5x - 2$. Therefore, the correct answer is indeed Option 3: $\frac{3}{5}$.

Question

If  $	ext{If } \left(\frac{5}{7}ight)^4 	imes \left(\frac{5}{7}ight)^{-3} = \left(\frac{5}{7}ight)^{5x-2}$, then x is-

Accepted Answer

To solve the given equation $\left(\frac{5}{7}\right)^4 \times \left(\frac{5}{7}\right)^{-3} = \left(\frac{5}{7}\right)^{5x-2}$, we need to simplify the left-hand side and equate the exponents.

### Step-by-Step Explanation:

1. **Simplify the Left-Hand Side:**
   - The expression $\left(\frac{5}{7}\right)^4 \times \left(\frac{5}{7}\right)^{-3}$ can be simplified using the property of exponents: $a^m \times a^n = a^{m+n}$.
   - Applying this property, we get:
     $$
     \left(\frac{5}{7}\right)^4 \times \left(\frac{5}{7}\right)^{-3} = \left(\frac{5}{7}\right)^{4 + (-3)} = \left(\frac{5}{7}\right)^1
     $$

2. **Equate the Exponents:**
   - Now, we have $\left(\frac{5}{7}\right)^1 = \left(\frac{5}{7}\right)^{5x-2}$.
   - Since the bases are the same, we can equate the exponents:
     $$
     1 = 5x - 2
     $$

3. **Solve for $x$:**
   - Add 2 to both sides of the equation:
     $$
     1 + 2 = 5x
     $$
     $$
     3 = 5x
     $$
   - Divide both sides by 5 to solve for $x$:
     $$
     x = \frac{3}{5}
     $$

### Conclusion:

- The value of $x$ is $\frac{3}{5}$, which corresponds to Option 3.

### Explanation of Incorrect Options:

1. **Option 1: $\frac{1}{5}$**
   - If $x = \frac{1}{5}$, then $5x = 1$, and $5x - 2 = 1 - 2 = -1$, which does not satisfy the equation $1 = 5x - 2$.

2. **Option 2: $\frac{2}{5}$**
   - If $x = \frac{2}{5}$, then $5x = 2$, and $5x - 2 = 2 - 2 = 0$, which does not satisfy the equation $1 = 5x - 2$.

3. **Option 4: $\frac{4}{5}$**
   - If $x = \frac{4}{5}$, then $5x = 4$, and $5x - 2 = 4 - 2 = 2$, which does not satisfy the equation $1 = 5x - 2$.

Therefore, the correct answer is indeed Option 3: $\frac{3}{5}$.