Simplify (ax3/2x)³: Cube Power of an Algebraic Fraction

Insert the corresponding expression:

$\left(\frac{a\times3}{2\times x}\right)^3=$

To solve this problem, we'll follow these steps:

• Step 1: Identify the given expression and its structure.
• Step 2: Apply the rule for powers of a fraction, $\left(\frac{p}{q}\right)^n$.
• Step 3: Calculate powers of individual components within the fraction.
• Step 4: Simplify the resulting expression.

Now, let's work through each step:

Step 1: Our given expression is $\left(\frac{a \times 3}{2 \times x}\right)^3$.

Step 2: Apply the power to the entire fraction using $\left(\frac{p}{q}\right)^n = \frac{p^n}{q^n}$, we get:
$\left(\frac{a \times 3}{2 \times x}\right)^3 = \frac{(a \times 3)^3}{(2 \times x)^3}$.

Step 3: Simplify the numerator and the denominator separately:
Numerator: $(a \times 3)^3 = a^3 \times 3^3 = a^3 \times 27$.
Denominator: $(2 \times x)^3 = 2^3 \times x^3 = 8 \times x^3$.

Step 4: Combine the simplified components to form the final expression:
The expression is $\frac{a^3 \times 27}{8 \times x^3}$.

Therefore, the solution to the problem is $\frac{a^3 \times 27}{8 \times x^3}$, which corresponds to choice 2.