Compare Complex Expressions: ((x×y)⁵)¹² vs ((x×y)¹⁰)²)³

To solve the problem, we need to simplify and compare the following two expressions:

$\left(\left(x \times y\right)^5\right)^{12} \quad \text{and} \quad \left(\left(\left(x \times y\right)^{10}\right)^2\right)^3$

Let's simplify each expression:

• The first expression $\left(\left(x \times y\right)^5\right)^{12}$.
  Using the power of a power rule, $\left(a^m\right)^n = a^{m \cdot n}$, we have:
\end{ul>

$\left(\left(x \times y\right)^5\right)^{12} = \left(x \times y\right)^{5 \cdot 12} = \left(x \times y\right)^{60}$

• The second expression $\left(\left(\left(x \times y\right)^{10}\right)^2\right)^3$.
  Similarly, apply the power of a power rule twice:

$\left(\left(\left(x \times y\right)^{10}\right)^2\right)^3 = \left(\left(x \times y\right)^{10 \cdot 2}\right)^3$

$= \left(x \times y\right)^{20 \cdot 3} = \left(x \times y\right)^{60}$

After simplification, both expressions become $\left(x \times y\right)^{60}$.

Therefore, the relationship between the two expressions is:

$\left(x \times y\right)^{60} = \left(x \times y\right)^{60}$

=

Thus, the correct choice from the provided options is:

: =

I am confident that the solution is correct, as both expressions simplify to the same value.