Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (64^5/3)/(64^4/3)

Verified step by step guidance

1. Identify the expression to simplify: \( \frac{64^{5/3}}{64^{4/3}} \).
2. Apply the quotient rule for exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
3. Subtract the exponents: \( 64^{5/3 - 4/3} \).
4. Simplify the exponent: \( 64^{1/3} \).
5. Recognize that \( 64^{1/3} \) is the cube root of 64.

Key Concepts

Exponential Rules

Exponential rules govern how to manipulate expressions involving exponents. 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.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the opposite positive exponent (a^(-n) = 1/a^n). In the context of simplification, it is important to rewrite expressions to eliminate negative exponents, ensuring that all variables and bases are expressed in a positive exponent form, as specified in the problem.

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 applying the rules of exponents and ensuring that the final expression adheres to any given constraints, such as avoiding negative exponents and maintaining positive values for variables.