Reduce (5²×2³×3)³×3² ÷ (2⁴×5³): Complex Exponential Expression

Let's reduce the given expression step-by-step using the laws of exponents.

The expression to simplify is:

$\frac{\left(5^2\times2^3\times3\right)^3\times3^2}{2^4\times5^3}$

First, simplify the expression inside the bracket:

• Apply the power of a power rule $(a^m)^n = a^{m \times n}$ to each term:

  • $(5^2)^3 = 5^{2 \times 3} = 5^6$

  • $(2^3)^3 = 2^{3 \times 3} = 2^9$

  • $(3^1)^3 = 3^{1 \times 3} = 3^3$

• Substitute back into the expression:

• $\frac{5^6 \times 2^9 \times 3^3 \times 3^2}{2^4 \times 5^3}$

Next, combine the powers in the numerator:

• Use the product of powers rule $a^m \times a^n = a^{m+n}$:

  • Combine the 3s: $3^3 \times 3^2 = 3^{3+2} = 3^5$

• The refined numerator is $5^6 \times 2^9 \times 3^5$.

Now, simplify the fraction using division of powers:

• For 5's: $\frac{5^6}{5^3} = 5^{6-3} = 5^3$

• For 2's: $\frac{2^9}{2^4} = 2^{9-4} = 2^5$

• 3's remain $3^5$, as they only appear in the numerator.

Therefore, the final expression is:

$\boxed{5^3 \times 2^5 \times 3^5}$