Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (27/64)^-4/3

Verified step by step guidance

1. Recognize that the expression is a power raised to another power: \(\left( \frac{27}{64} \right)^{-\frac{4}{3}}\). Use the property of exponents: \(\left( a^m \right)^n = a^{m \cdot n}\), which means you can rewrite the expression as \(\left( \frac{27}{64} \right)^{-\frac{4}{3}}\) without change, but this helps us think about simplifying the base first.
2. Rewrite the base numbers as powers of their prime factors: \$27 = 3^3\( and \)64 = 4^3 = 2^6\(, but since \)64 = 2^6\( is not a perfect cube, let's use \)64 = 4^3\( is incorrect. Actually, \)64 = 2^6\(, but since the exponent is \)-\frac{4}{3}\(, it's easier to write \)64 = 4^3\( is incorrect. Instead, write \)64 = 4^3\( is wrong; the correct prime factorization is \)64 = 2^6\(. So, rewrite the fraction as \)\left( \frac{3^3}{2^6} \right)^{-\frac{4}{3}}$.
3. Apply the exponent to both numerator and denominator separately: \(\left( 3^3 \right)^{-\frac{4}{3}}\) and \(\left( 2^6 \right)^{-\frac{4}{3}}\). Use the power of a power rule: \(\left( a^m \right)^n = a^{m \cdot n}\), so the numerator becomes \$3^{3 \times -\frac{4}{3}}\( and the denominator becomes \)2^{6 \times -\frac{4}{3}}$.
4. Simplify the exponents by multiplying: For the numerator, \$3 \times -\frac{4}{3} = -4\(, so the numerator is \)3^{-4}\(. For the denominator, \)6 \times -\frac{4}{3} = -8\(, so the denominator is \)2^{-8}\(. The expression is now \)\frac{3^{-4}}{2^{-8}}$.
5. Rewrite the expression to eliminate negative exponents by using the rule \(a^{-m} = \frac{1}{a^m}\). So, \$3^{-4} = \frac{1}{3^4}\( and \)2^{-8} = \frac{1}{2^8}\(. Since the denominator has a negative exponent, it moves to the numerator as a positive exponent. Therefore, the expression becomes \)\frac{1}{3^4} \times 2^8\(, which can be written as \)\frac{2^8}{3^4}$.

Key Concepts

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. For example, a^(-n) = 1/a^n. Simplifying expressions with negative exponents often involves rewriting them without negatives by taking reciprocals.

Rational Exponents

Rational exponents represent roots and powers simultaneously. An expression like a^(m/n) means the nth root of a raised to the mth power, or (√[n]{a})^m. Understanding this helps simplify expressions involving fractional powers.

Properties of Exponents and Radicals

Exponent rules such as (a^m)^n = a^(mn) and (a/b)^m = a^m / b^m allow simplification of complex expressions. Recognizing how to apply these properties to both numerator and denominator is essential for rewriting expressions without negative exponents.