Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (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. Recall the property of exponents for division: \(\frac{a^m}{a^n} = a^{m-n}\), where \(a\) is a positive real number.
3. Apply the exponent subtraction rule to the expression: \$64^{5/3 - 4/3}$.
4. Simplify the exponent by subtracting the fractions: \$5/3 - 4/3 = 1/3\(, so the expression becomes \)64^{1/3}$.
5. Rewrite \$64^{1/3}\( as the cube root of 64, which is \)\sqrt[3]{64}$, and express the answer without negative exponents.

Key Concepts

Properties of Exponents

Exponents follow specific rules such as when dividing like bases, subtract the exponents (a^m / a^n = a^(m-n)). Understanding these properties allows simplification of expressions involving powers efficiently.

Fractional Exponents

Fractional exponents represent roots and powers simultaneously; for example, a^(m/n) means the nth root of a raised to the mth power. Recognizing this helps in rewriting and simplifying expressions involving radicals and powers.

Eliminating Negative Exponents

Negative exponents indicate reciprocals (a^(-n) = 1/a^n). To write answers without negative exponents, rewrite expressions by moving factors with negative exponents between numerator and denominator.