Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (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 rule \((a/b)^m = a^m / b^m\) to rewrite the expression as \(\frac{27^{-\frac{4}{3}}}{64^{-\frac{4}{3}}}\).
2. Apply the negative exponent rule \(a^{-m} = \frac{1}{a^m}\) to rewrite the expression as \(\frac{1}{27^{\frac{4}{3}}} \div \frac{1}{64^{\frac{4}{3}}}\), which simplifies to \(\frac{64^{\frac{4}{3}}}{27^{\frac{4}{3}}}\).
3. Rewrite the bases 27 and 64 as powers of prime numbers: \$27 = 3^3\( and \)64 = 4^3\( or more precisely \)64 = 2^6\(, but since \)64 = 4^3\( is incorrect, use \)64 = 2^6$ for clarity.
4. Substitute the prime factorizations into the expression: \(\frac{(2^6)^{\frac{4}{3}}}{(3^3)^{\frac{4}{3}}}\). Use the power of a power rule \((a^m)^n = a^{mn}\) to simplify the exponents.
5. Calculate the new exponents by multiplying: numerator exponent \$6 \times \frac{4}{3} = 8\(, denominator exponent \)3 \times \frac{4}{3} = 4\(. So the expression becomes \)\frac{2^8}{3^4}$. This is the simplified form without negative exponents.

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 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 n-th root of a raised to the m-th power, or (√[n]{a})^m. Understanding how to manipulate these helps simplify expressions involving fractional powers.

Properties of Exponents and Radicals

Exponent rules such as (a^m)^n = a^(mn) and (a/b)^n = a^n / b^n allow simplification of complex expressions. Recognizing how to apply these properties, especially with roots and powers, is essential for rewriting expressions in simpler forms without negative exponents.