If x is not equal to 0, what is the value of `(6(2x)^(-2))/((3x)^(-2))`?

Answer

To solve the expression \((6(2x)^{-2})/((3x)^{-2})\), we will follow these steps:

### Step 1: Rewrite the expression
The expression can be rewritten using the property of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\). Thus, we can rewrite the expression as:
\[ \frac{6}{(2x)^2} \div \frac{1}{(3x)^2} \]

### Step 2: Simplify the division
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as:
\[ 6 \cdot \frac{(3x)^2}{(2x)^2} \]

### Step 3: Calculate the squares
Now we will calculate the squares:
\[ (3x)^2 = 9x^2 \quad \text{and} \quad (2x)^2 = 4x^2 \]
Substituting these values back into the expression gives us:
\[ 6 \cdot \frac{9x^2}{4x^2} \]

### Step 4: Simplify the fraction
The \(x^2\) terms in the numerator and denominator cancel out:
\[ 6 \cdot \frac{9}{4} \]

### Step 5: Multiply the constants
Now we multiply the constants:
\[ 6 \cdot \frac{9}{4} = \frac{54}{4} \]

### Step 6: Simplify the fraction
Finally, we simplify \(\frac{54}{4}\) by dividing both the numerator and the denominator by 2:
\[ \frac{54 \div 2}{4 \div 2} = \frac{27}{2} \]

Thus, the value of the expression \((6(2x)^{-2})/((3x)^{-2})\) is:
\[ \frac{27}{2} \]

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