Multiply or divide, as indicated. See Example 3. (15p³ / 9p²) • (12p / 10p³)

Verified step by step guidance

1. Rewrite the expression clearly as a multiplication of two fractions: \(\frac{15p^{3}}{9p^{2}} \times \frac{12p}{10p^{3}}\).
2. Simplify each fraction separately by dividing the coefficients (numerical parts) and applying the laws of exponents to the variable parts. Recall that \(\frac{p^{a}}{p^{b}} = p^{a-b}\).
3. For the first fraction, simplify \(\frac{15}{9}\) by dividing numerator and denominator by their greatest common divisor, and simplify \(\frac{p^{3}}{p^{2}}\) by subtracting exponents.
4. For the second fraction, simplify \(\frac{12}{10}\) similarly, and simplify \(\frac{p}{p^{3}}\) by subtracting exponents.
5. After simplifying both fractions, multiply the simplified coefficients and multiply the powers of \(p\) by adding their exponents according to the multiplication rule \(p^{m} \times p^{n} = p^{m+n}\).

Key Concepts

Multiplication and Division of Algebraic Fractions

When multiplying or dividing algebraic fractions, multiply or divide the numerators and denominators separately. For division, multiply by the reciprocal of the divisor. This process simplifies complex expressions involving variables and coefficients.

Laws of Exponents

Exponents follow specific rules: when multiplying like bases, add the exponents; when dividing, subtract the exponents. For example, p³ × p = p^(3+1) = p⁴, and p³ ÷ p² = p^(3-2) = p¹. These laws help simplify expressions with powers.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involves canceling common factors in numerators and denominators and combining like terms. This reduces the expression to its simplest form, making it easier to interpret or use in further calculations.