Compare Algebraic Fractions: Evaluating (ab/xy)¹⁰ vs Complex Fraction Expression

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

• Step 1: Simplify both expressions using the properties of exponents.
• Step 2: Compare the simplified results to determine the correct sign.

Now, let us simplify each side:

Step 1: Simplifying the left-hand side:
$\frac{a^{10} \times b^{10}}{x^{10} \times y^{10}}$

This expression can be expanded as:
$\frac{a^{10} b^{10}}{x^{10} y^{10}} = \left(\frac{a^{10}}{x^{10}}\right) \times \left(\frac{b^{10}}{y^{10}}\right)$
$= \left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10}$

Step 2: Simplifying the right-hand side:
$\left(\frac{a \times b}{x \times y}\right)^{10}$

This can be expressed as:
$\left( \frac{a \times b}{x \times y} \right)^{10} = \left( \frac{a}{x} \times \frac{b}{y} \right)^{10}$
With the rule of exponentiating a product, this becomes:
$= \left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10}$

After simplifying both expressions, we can see:
Both the left-hand side and the right-hand side are identical:

$\left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10} = \left(\frac{a}{x}\right)^{10} \times \left(\frac{b}{y}\right)^{10}$

Therefore, the correct sign to use is $=$.

The comparative relationship between the expressions is equal.

Thus, the solution to the problem is $=$.