Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (64^5/3)/(64^4/3)

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Identify the expression to simplify: \(\frac{64^{5/3}}{64^{4/3}}\).

Recall the property of exponents for division: \(\frac{a^m}{a^n} = a^{m-n}\), where \(a\) is a positive real number.

Apply the exponent subtraction rule to the expression: \(64^{5/3 - 4/3}\).

Simplify the exponent by subtracting the fractions: \(5/3 - 4/3 = 1/3\), so the expression becomes \(64^{1/3}\).

Rewrite \(64^{1/3}\) as the cube root of 64, which is \(\sqrt[3]{64}\), and express the answer without negative exponents.

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주요 개념

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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.

Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (645/3)/(644/3)