Simplify the Expression: (5×8)^(3+5y) ÷ (8×5)^(3y+1)

Let's begin by examining the given expression: $\frac{\left(5\times8\right)^{3+5y}}{\left(8\times5\right)^{3y+1}}=$

Both the numerator and the denominator share the same base, $5 \times 8$, which can be expressed as $(40)$.

Next, we apply the quotient rule for exponents, which states that $\frac{a^m}{a^n} = a^{m-n}$, provided that $a \neq 0$.

We have:

• Numerator exponent: $(3 + 5y)$
• Denominator exponent: $(3y + 1)$

By applying the quotient rule, we can subtract the exponent in the denominator from the exponent in the numerator:

$(3 + 5y) - (3y + 1) = 3 + 5y - 3y - 1$

Simplifying the expression, we get:

• $3 - 1 = 2$
• $5y - 3y = 2y$

Combining these, we have:

$(40)^{2y + 2}$

Thus, the simplified form of the expression is:

$(5 \times 8)^{2y + 2}$

The solution to the question is: $(5 \times 8)^{2y + 2}$