If `root(m)(27) = 3^(3m) and 4^(m) > 1`, then what is the value of m?

Options

A. `-1`
B. `-1//4`
C. `0`
D. `1`

Answer

To solve the equation \( \sqrt[m]{27} = 3^{3m} \) under the condition \( 4^m > 1 \), we can follow these steps:

### Step 1: Analyze the inequality \( 4^m > 1 \)
We know that \( 4^m > 1 \) implies:
\[ m > 0 \]
This is because any positive base raised to a power greater than zero is greater than 1.

### Step 2: Rewrite the m-th root of 27
The m-th root of 27 can be expressed as:
\[ \sqrt[m]{27} = 27^{\frac{1}{m}} \]
Now, we can rewrite 27 in terms of base 3:
\[ 27 = 3^3 \]
Thus, we have:
\[ \sqrt[m]{27} = (3^3)^{\frac{1}{m}} = 3^{\frac{3}{m}} \]

### Step 3: Set the equation
Now we can set the equation from the problem:
\[ 3^{\frac{3}{m}} = 3^{3m} \]

### Step 4: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ \frac{3}{m} = 3m \]

### Step 5: Solve for m
To solve for \( m \), we can multiply both sides by \( m \) (noting that \( m > 0 \)):
\[ 3 = 3m^2 \]
Dividing both sides by 3 gives:
\[ 1 = m^2 \]

### Step 6: Find the values of m
Taking the square root of both sides, we find:
\[ m = 1 \quad \text{or} \quad m = -1 \]
However, since we established earlier that \( m > 0 \), we discard \( m = -1 \).

### Conclusion
Thus, the only solution is:
\[ m = 1 \]

### Final Answer
The value of \( m \) is \( 1 \).

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