The value if $(\frac{1}{64})^{-5/6}$ will be

Q: The value if $(\frac{1}{64})^{-5/6}$ will be

To solve the expression $(\frac{1}{64})^{-5/6}$, we need to understand the mathematical operations involved, particularly dealing with negative exponents and fractional exponents. ### Step-by-Step Explanation: 1. **Understanding Negative Exponents:** A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. Therefore, $(\frac{1}{64})^{-5/6}$ can be rewritten as $(64)^{5/6}$. 2. **Understanding Fractional Exponents:** A fractional exponent such as $5/6$ implies a root and a power. Specifically, $a^{m/n} = (a^{1/n})^m$, which means you first take the nth root of $a$ and then raise the result to the mth power. 3. **Applying the Fractional Exponent:** - First, find the 6th root of 64. Since $64 = 2^6$, the 6th root of 64 is $2$. - Next, raise the result to the power of 5: $2^5 = 32$. Thus, $(64)^{5/6} = 32$. ### Justification for Option 3: - **Option 3 (32) is correct** because, as shown above, $(\frac{1}{64})^{-5/6}$ simplifies to 32 through the process of taking the reciprocal, finding the 6th root, and then raising to the 5th power. ### Explanation for Incorrect Options: - **Option 1 (8):** This would imply that $(64)^{5/6}$ equals 8, which is incorrect because the 6th root of 64 is 2, and $2^5$ is not 8. - **Option 2 (16):** This would suggest that $(64)^{5/6}$ equals 16, which is incorrect because, as calculated, $2^5 = 32$, not 16. - **Option 4 (1/32):** This option is incorrect because it suggests a misunderstanding of the negative exponent. The negative exponent was already addressed by taking the reciprocal, and the correct calculation leads to 32, not its reciprocal. In conclusion, the correct answer is Option 3 (32), as it accurately reflects the value of $(\frac{1}{64})^{-5/6}$ through proper application of exponent rules.

Question

Accepted Answer

To solve the expression $(\frac{1}{64})^{-5/6}$, we need to understand the mathematical operations involved, particularly dealing with negative exponents and fractional exponents.

### Step-by-Step Explanation:

1. **Understanding Negative Exponents:**
   A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. Therefore, $(\frac{1}{64})^{-5/6}$ can be rewritten as $(64)^{5/6}$.

2. **Understanding Fractional Exponents:**
   A fractional exponent such as $5/6$ implies a root and a power. Specifically, $a^{m/n} = (a^{1/n})^m$, which means you first take the nth root of $a$ and then raise the result to the mth power.

3. **Applying the Fractional Exponent:**
   - First, find the 6th root of 64. Since $64 = 2^6$, the 6th root of 64 is $2$.
   - Next, raise the result to the power of 5: $2^5 = 32$.

Thus, $(64)^{5/6} = 32$.

### Justification for Option 3:

- **Option 3 (32) is correct** because, as shown above, $(\frac{1}{64})^{-5/6}$ simplifies to 32 through the process of taking the reciprocal, finding the 6th root, and then raising to the 5th power.

### Explanation for Incorrect Options:

- **Option 1 (8):** This would imply that $(64)^{5/6}$ equals 8, which is incorrect because the 6th root of 64 is 2, and $2^5$ is not 8.

- **Option 2 (16):** This would suggest that $(64)^{5/6}$ equals 16, which is incorrect because, as calculated, $2^5 = 32$, not 16.

- **Option 4 (1/32):** This option is incorrect because it suggests a misunderstanding of the negative exponent. The negative exponent was already addressed by taking the reciprocal, and the correct calculation leads to 32, not its reciprocal.

In conclusion, the correct answer is Option 3 (32), as it accurately reflects the value of $(\frac{1}{64})^{-5/6}$ through proper application of exponent rules.