Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. -81^3/4

Verified step by step guidance

1. Recognize that the expression is \(-81^{3/4}\), which means the negative sign is outside the exponentiation. So, first focus on simplifying \$81^{3/4}$.
2. Rewrite 81 as a power of a prime number: \$81 = 3^4\(. Substitute this into the expression to get \)(3^4)^{3/4}$.
3. Use the power of a power property: \((a^m)^n = a^{m \times n}\). Multiply the exponents: \$4 \times \frac{3}{4} = 3\(, so the expression becomes \)3^3$.
4. Calculate \$3^3\( by multiplying 3 by itself three times: \)3 \times 3 \times 3$ (do not compute the final value, just express it this way).
5. Finally, apply the negative sign outside the expression to get \(-3^3\). This is the simplified form without negative exponents.

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, i.e., (√[n]{a})^m. Understanding how to manipulate these helps simplify expressions involving fractional powers.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent, such as a^(-n) = 1/a^n. Simplifying expressions often requires rewriting negative exponents as positive by moving terms between numerator and denominator.

Simplifying Expressions with Radicals and Exponents

Simplifying expressions with radicals and exponents involves converting between radical and exponential forms, applying exponent rules, and combining like terms. This process ensures the final answer is in simplest form without negative exponents, as required.