Example problem demonstrating simplification of algebraic expressions using exponent rules.

Problem:

The task is to simplify the given expressions using exponent rules. The expressions are:

1. \( \frac{(3x^2y^7)(6x^4y^3)}{2xy^4} \)
2. \( \frac{(a^2b^5)(a^3b^7)}{(ab^2)^3} \)

We will solve each part step by step.

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Part (a): Simplify \( \frac{(3x^2y^7)(6x^4y^3)}{2xy^4} \)

#### Step 1: Multiply the numerator
The numerator is \( (3x^2y^7)(6x^4y^3) \). Using the properties of exponents, we multiply the coefficients and add the exponents of like bases:

\[
(3x^2y^7)(6x^4y^3) = (3 \cdot 6)(x^{2+4})(y^{7+3}) = 18x^6y^{10}
\]

So, the expression becomes:

\[
\frac{18x^6y^{10}}{2xy^4}
\]

#### Step 2: Divide by the denominator
Now, divide the numerator by the denominator \( 2xy^4 \):

- Divide the coefficients: \( \frac{18}{2} = 9 \)
- Subtract the exponents of like bases in the numerator and denominator:
- For \( x \): \( x^6 \div x^1 = x^{6-1} = x^5 \)
- For \( y \): \( y^{10} \div y^4 = y^{10-4} = y^6 \)

Thus, the simplified expression is:

\[
9x^5y^6
\]

#### Final Answer for Part (a):
\[
\boxed{9x^5y^6}
\]

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Part (b): Simplify \( \frac{(a^2b^5)(a^3b^7)}{(ab^2)^3} \)

#### Step 1: Simplify the numerator
The numerator is \( (a^2b^5)(a^3b^7) \). Using the properties of exponents, we multiply the terms:

\[
(a^2b^5)(a^3b^7) = a^{2+3}b^{5+7} = a^5b^{12}
\]

So, the expression becomes:

\[
\frac{a^5b^{12}}{(ab^2)^3}
\]

#### Step 2: Simplify the denominator
The denominator is \( (ab^2)^3 \). Using the power of a product rule \( (xy)^n = x^n y^n \), we distribute the exponent 3:

\[
(ab^2)^3 = a^3(b^2)^3 = a^3b^{2 \cdot 3} = a^3b^6
\]

So, the expression becomes:

\[
\frac{a^5b^{12}}{a^3b^6}
\]

#### Step 3: Divide the numerator by the denominator
Now, divide the numerator by the denominator:

- For \( a \): \( a^5 \div a^3 = a^{5-3} = a^2 \)
- For \( b \): \( b^{12} \div b^6 = b^{12-6} = b^6 \)

Thus, the simplified expression is:

\[
a^2b^6
\]

#### Final Answer for Part (b):
\[
\boxed{a^2b^6}
\]

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Summary of Answers:

1. \( \frac{(3x^2y^7)(6x^4y^3)}{2xy^4} = \boxed{9x^5y^6} \)
2. \( \frac{(a^2b^5)(a^3b^7)}{(ab^2)^3} = \boxed{a^2b^6} \)