Determine the value of 'x' when `(root(3)(3))^x` = 27

Options

A. 1
B. 3
C. 9
D. 12

Answer

To solve the equation \((\sqrt[3]{3})^x = 27\), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ (\sqrt[3]{3})^x = 27 \]
We know that \(27\) can be expressed as a power of \(3\):
\[ 27 = 3^3 \]
So we can rewrite the equation as:
\[ (\sqrt[3]{3})^x = 3^3 \]

### Step 2: Rewrite the left side using exponents
The cube root of \(3\) can be expressed as:
\[ \sqrt[3]{3} = 3^{1/3} \]
Thus, we can rewrite the left side of the equation:
\[ (3^{1/3})^x = 3^3 \]

### Step 3: Apply the power of a power rule
Using the power of a power property \((a^m)^n = a^{m \cdot n}\), we can simplify the left side:
\[ 3^{x/3} = 3^3 \]

### Step 4: Set the exponents equal to each other
Since the bases are the same, we can set the exponents equal to each other:
\[ \frac{x}{3} = 3 \]

### Step 5: Solve for \(x\)
To solve for \(x\), multiply both sides by \(3\):
\[ x = 3 \cdot 3 \]
\[ x = 9 \]

Thus, the value of \(x\) is:
\[ \boxed{9} \]

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