If $a^x = b, b^y = c and c^z$ = a, then what is the value of xyz ?

Q: If $a^x = b, b^y = c and c^z$ = a, then what is the value of xyz ?

To determine the value of $ xyz $ given the equations $ a^x = b $, $ b^y = c $, and $ c^z = a $, we need to analyze the relationships between the variables. 1. **Analyzing the Equations:** - From $ a^x = b $, we can express $ a $ in terms of $ b $ as $ a = b^{1/x} $. - From $ b^y = c $, we can express $ b $ in terms of $ c $ as $ b = c^{1/y} $. - From $ c^z = a $, we can express $ c $ in terms of $ a $ as $ c = a^{1/z} $. 2. **Substituting and Simplifying:** - Substitute $ b = c^{1/y} $ into $ a = b^{1/x} $: \[ a = (c^{1/y})^{1/x} = c^{1/(xy)} \] - Substitute $ c = a^{1/z} $ into the above expression: \[ a = (a^{1/z})^{1/(xy)} = a^{1/(xyz)} \] 3. **Equating Powers:** - Since $ a = a^{1/(xyz)} $, for this equality to hold for all $ a \neq 0 $, the exponent must be 1: \[ 1 = \frac{1}{xyz} \] - Solving this gives: \[ xyz = 1 \] Thus, the value of $ xyz $ is indeed 1, which corresponds to Option 2. 3. **Explanation of Incorrect Options:** - **Option 1 (abc):** This option suggests that $ xyz = abc $. However, our derivation shows that $ xyz = 1 $, not dependent on the values of $ a $, $ b $, or $ c $. - **Option 3 ($\frac{1}{abc}$):** This option implies that $ xyz = \frac{1}{abc} $. Again, our derivation shows that $ xyz = 1 $, which is independent of $ a $, $ b $, or $ c $. - **Option 4 (ab):** This option suggests that $ xyz = ab $. Our derivation clearly shows that $ xyz = 1 $, which does not involve the product $ ab $. In conclusion, the correct value of $ xyz $ is 1, which is why Option 2 is the correct answer.

Question

Accepted Answer

To determine the value of $ xyz $ given the equations $ a^x = b $, $ b^y = c $, and $ c^z = a $, we need to analyze the relationships between the variables.

1. **Analyzing the Equations:**

- From $ a^x = b $, we can express $ a $ in terms of $ b $ as $ a = b^{1/x} $.
   - From $ b^y = c $, we can express $ b $ in terms of $ c $ as $ b = c^{1/y} $.
   - From $ c^z = a $, we can express $ c $ in terms of $ a $ as $ c = a^{1/z} $.

2. **Substituting and Simplifying:**

- Substitute $ b = c^{1/y} $ into $ a = b^{1/x} $:
     $$
     a = (c^{1/y})^{1/x} = c^{1/(xy)}
     $$
   - Substitute $ c = a^{1/z} $ into the above expression:
     $$
     a = (a^{1/z})^{1/(xy)} = a^{1/(xyz)}
     $$

3. **Equating Powers:**

- Since $ a = a^{1/(xyz)} $, for this equality to hold for all $ a \neq 0 $, the exponent must be 1:
     $$
     1 = \frac{1}{xyz}
     $$
   - Solving this gives:
     $$
     xyz = 1
     $$

Thus, the value of $ xyz $ is indeed 1, which corresponds to Option 2.

3. **Explanation of Incorrect Options:**

- **Option 1 (abc):** This option suggests that $ xyz = abc $. However, our derivation shows that $ xyz = 1 $, not dependent on the values of $ a $, $ b $, or $ c $.
   
   - **Option 3 ($\frac{1}{abc}$):** This option implies that $ xyz = \frac{1}{abc} $. Again, our derivation shows that $ xyz = 1 $, which is independent of $ a $, $ b $, or $ c $.
   
   - **Option 4 (ab):** This option suggests that $ xyz = ab $. Our derivation clearly shows that $ xyz = 1 $, which does not involve the product $ ab $.

In conclusion, the correct value of $ xyz $ is 1, which is why Option 2 is the correct answer.