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

Verified step by step guidance

1. Rewrite the expression clearly as \(\left( \frac{-4m^{2}}{tp^{2}} \right)^{4}\) to understand the structure.
2. Apply the exponent of 4 to both the numerator and the denominator separately, using the property \(\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}\), so it becomes \(\frac{(-4m^{2})^{4}}{(tp^{2})^{4}}\).
3. Simplify the numerator by raising each factor to the 4th power: \((-4)^{4}\) and \((m^{2})^{4}\). Use the power of a power rule \(\left(a^{m}\right)^{n} = a^{mn}\) to get \(m^{8}\).
4. Simplify the denominator by raising each factor to the 4th power: \(t^{4}\) and \((p^{2})^{4} = p^{8}\).
5. Combine the simplified numerator and denominator to write the final simplified expression as \(\frac{(-4)^{4} m^{8}}{t^{4} p^{8}}\).

Key Concepts

Exponentiation of Fractions

When raising a fraction to a power, both the numerator and denominator are raised to that power separately. For example, (a/b)^n = a^n / b^n. This rule helps simplify expressions involving powers of fractions.

Laws of Exponents

The laws of exponents govern how to handle powers of variables, such as (x^m)^n = x^(m*n) and (xy)^n = x^n * y^n. These rules allow simplification of expressions with variables raised to powers.

Handling Negative and Nonzero Variables

Assuming variables are nonzero ensures division is valid and avoids undefined expressions. Recognizing the sign and domain of variables helps correctly simplify expressions, especially when dealing with even powers that affect sign.