The problem asks to simplify the expression $[5a^5b^2 \times 3(ab^3)^2] \div (15a^2b^8)$.

1. Problem Description

The problem asks to simplify the expression $[5a^5b^2 \times 3(ab^3)^2] \div (15a^2b^8)$.

2. Solution Steps

First, simplify the term $(ab^3)^2$ using the power of a product rule: $(xy)^n = x^n y^n$.

$(ab^3)^2 = a^2(b^3)^2 = a^2b^6$

Now substitute this back into the original expression:

$[5a^5b^2 \times 3a^2b^6] \div (15a^2b^8)$

Next, simplify the expression inside the brackets by multiplying the terms:

$5a^5b^2 \times 3a^2b^6 = (5 \times 3)(a^5 \times a^2)(b^2 \times b^6)$

Using the product of powers rule, $x^m \times x^n = x^{m+n}$:

$15a^{5+2}b^{2+6} = 15a^7b^8$

Substitute this back into the expression:

$[15a^7b^8] \div (15a^2b^8)$

Now, divide the two terms:

$\frac{15a^7b^8}{15a^2b^8}$

$\frac{15}{15} \times \frac{a^7}{a^2} \times \frac{b^8}{b^8}$

$1 \times \frac{a^7}{a^2} \times \frac{b^8}{b^8}$

Using the quotient of powers rule, $\frac{x^m}{x^n} = x^{m-n}$:

$a^{7-2}b^{8-8} = a^5b^0$

Since $b^0 = 1$, we have:

$a^5 \times 1 = a^5$

3. Final Answer

$a^5$