Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. -81^3/4

Verified step by step guidance

1. Recognize that the expression is \(-81^{3/4}\), which means \(-81\) raised to the power of \(\frac{3}{4}\).
2. Rewrite the base \(-81\) as \(-1 \times 81\) to separate the negative sign and the positive number: \((-1) \times 81\).
3. Express \(81\) as a power of a positive base: since \(81 = 3^4\), rewrite the expression as \((-1) \times (3^4)^{3/4}\).
4. Apply the power of a power rule \( (a^m)^n = a^{m \times n} \) to simplify \((3^4)^{3/4} = 3^{4 \times \frac{3}{4}} = 3^3\).
5. Now the expression is \((-1)^{3/4} \times 3^3\). Since the problem assumes all variables represent positive real numbers and to avoid complex numbers, consider the absolute value and rewrite the expression accordingly to eliminate negative exponents and simplify.

Key Concepts

Rational Exponents

Rational exponents represent roots and powers simultaneously. For example, a^(m/n) means the nth root of a raised to the mth power. Understanding how to interpret and manipulate these exponents is essential for simplifying expressions like -81^(3/4).

Properties of Exponents

Properties such as a^(m) * a^(n) = a^(m+n) and (a^m)^n = a^(m*n) help simplify expressions with exponents. Applying these rules allows rewriting expressions to eliminate negative exponents and combine powers effectively.

Simplifying Negative Bases with Rational Exponents

When dealing with negative bases raised to rational exponents, it's important to consider the domain and whether the expression is defined. Since variables are positive here, the negative sign is separate, and simplifying involves handling the positive base first, then applying the negative sign.