If `(729)^(2)= 9^(x)`, then find 5x.

Answer

To solve the equation \( (729)^2 = 9^x \), we will follow these steps:

### Step 1: Rewrite 729 in terms of base 9
First, we need to express 729 as a power of 9. We know that:
\[ 729 = 9^3 \]
This can be verified since \( 9 \times 9 \times 9 = 729 \).

### Step 2: Substitute 729 in the equation
Now, we can substitute \( 729 \) in the original equation:
\[ (729)^2 = (9^3)^2 \]

### Step 3: Apply the power of a power property
Using the property of exponents that states \( (a^m)^n = a^{m \cdot n} \), we can simplify the left side:
\[ (9^3)^2 = 9^{3 \cdot 2} = 9^6 \]

### Step 4: Set the exponents equal to each other
Now we have:
\[ 9^6 = 9^x \]
Since the bases are the same, we can set the exponents equal to each other:
\[ 6 = x \]

### Step 5: Find \( 5x \)
Now that we have found \( x \), we can calculate \( 5x \):
\[ 5x = 5 \cdot 6 = 30 \]

### Final Answer
Thus, the value of \( 5x \) is:
\[ \boxed{30} \]

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