Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (3pq)q^2/6p^2q^4

Verified step by step guidance

1. Start by simplifying the expression $(3pq)q^2$. This can be rewritten as $3pq\cdot q^2$.
2. Apply the property of exponents $a^m\cdot a^n=a^{m+n}$ to combine the $q$ terms: $q\cdot q^2=q^{1+2}=q^3$.
3. Now the expression becomes $\frac{3p{q}^3}{6p^2{q}^4}$.
4. Simplify the coefficients: $\frac{3}{6}=\frac{1}{2}$.
5. Apply the property of exponents $\frac{a^m}{a^n}=a^{m-n}$ to both $p$ and $q$: $\frac{p}{p^2}=p^{1-2}=p^{-1}$ and $\frac{q^3}{q^4}=q^{3-4}=q^{-1}$.

Key Concepts

Exponent Rules

Exponent rules are fundamental principles that govern how to manipulate expressions involving powers. Key rules include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these rules is essential for simplifying expressions with exponents.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. This process often requires factoring polynomials and applying the rules of exponents to eliminate any negative exponents, ensuring that the final expression is presented in a standard format.

Nonzero Variables

In algebra, it is crucial to assume that variables represent nonzero real numbers when simplifying expressions. This assumption prevents division by zero, which is undefined. It also allows for the cancellation of variables in rational expressions without concern for invalid operations, ensuring that the simplification process remains valid.