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. Start by writing the expression clearly: \(\frac{64^{\frac{5}{3}}}{64^{\frac{4}{3}}}\).
2. Recall the property of exponents for division with the same base: \(\frac{a^m}{a^n} = a^{m-n}\). Apply this to get \$64^{\frac{5}{3} - \frac{4}{3}}$.
3. Subtract the exponents: \(\frac{5}{3} - \frac{4}{3} = \frac{1}{3}\), so the expression simplifies to \$64^{\frac{1}{3}}$.
4. Recognize that \$64^{\frac{1}{3}}\( means the cube root of 64, which can be written as \)\sqrt[3]{64}$.
5. Since 64 is a perfect cube (\$64 = 4^3\(), rewrite \)\sqrt[3]{64}$ as 4, completing the simplification without negative exponents.

Key Concepts

Properties of Exponents

This concept involves rules for manipulating expressions with exponents, such as multiplying powers with the same base by adding exponents, dividing by subtracting exponents, and raising a power to another power by multiplying exponents. Understanding these rules is essential for simplifying expressions like (64^(5/3)) / (64^(4/3)).

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. Recognizing how to interpret and simplify expressions with fractional exponents helps in rewriting and simplifying terms like 64^(5/3).

Simplifying Expressions Without Negative Exponents

This involves rewriting expressions so that no exponents are negative, often by using reciprocal properties of exponents. Since the problem specifies answers without negative exponents, it is important to convert any negative powers into positive ones by moving factors between numerator and denominator.