Simplify each expression. Assume all variables represent nonzero real numbers. See Examples 2 and 3. -4m²( ——— )⁴ tp²

Verified step by step guidance

1. Step 1: Recognize that the expression \( \left( \frac{-4m^2}{tp^2} \right)^4 \) involves raising a fraction to a power.
2. Step 2: Apply the power of a quotient rule, which states \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \).
3. Step 3: Raise the numerator \(-4m^2\) to the fourth power: \((-4m^2)^4\).
4. Step 4: Raise the denominator \(tp^2\) to the fourth power: \((tp^2)^4\).
5. Step 5: Simplify each part separately: \((-4)^4\), \((m^2)^4\), \(t^4\), and \((p^2)^4\), then combine the results.

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 power of a power ( (a^m)^n = a^(m*n)), and the power of a product ( (ab)^n = a^n * b^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 recognizing equivalent expressions. Mastery of this concept is crucial for effectively simplifying complex algebraic fractions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent (a^(-n) = 1/a^n). This concept is important when simplifying expressions, as it allows for the transformation of terms and can lead to a clearer representation of the expression. Recognizing and applying this rule is vital in the simplification process.