Simplify and express each of the following as a rational number :
` (3^(4) ) ^(2) `

Answer

To simplify the expression \( (3^4)^2 \) and express it as a rational number, we can follow these steps:

### Step 1: Apply the Power of a Power Rule
We know that when we have an exponent raised to another exponent, we can multiply the exponents. This is expressed mathematically as:
\[ a^{m^n} = a^{m \cdot n} \]
In our case, we have:
\[ (3^4)^2 = 3^{4 \cdot 2} \]

### Step 2: Calculate the Exponent
Now we will calculate the exponent:
\[ 4 \cdot 2 = 8 \]
So, we can rewrite the expression as:
\[ (3^4)^2 = 3^8 \]

### Step 3: Express as a Rational Number
Next, we need to express \( 3^8 \) as a rational number. Any integer can be expressed as a fraction by placing it over 1:
\[ 3^8 = \frac{3^8}{1} \]

### Step 4: Calculate \( 3^8 \)
Now, we can calculate \( 3^8 \):
\[ 3^8 = 6561 \]
Thus, we can express it as:
\[ \frac{6561}{1} \]

### Final Answer
The simplified expression of \( (3^4)^2 \) as a rational number is:
\[ \frac{6561}{1} \]

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