Evaluate each expression 27^(-4/3)

Verified step by step guidance

1. Rewrite the expression using the property of exponents: \( a^{-b} = \frac{1}{a^b} \). So, \( 27^{-\frac{4}{3}} = \frac{1}{27^{\frac{4}{3}}} \).
2. Recognize that \( 27 \) can be expressed as a power of 3: \( 27 = 3^3 \).
3. Substitute \( 27 = 3^3 \) into the expression: \( \frac{1}{(3^3)^{\frac{4}{3}}} \).
4. Apply the power of a power property: \((a^m)^n = a^{m \cdot n}\). So, \((3^3)^{\frac{4}{3}} = 3^{3 \cdot \frac{4}{3}} = 3^4\).
5. Simplify the expression: \( \frac{1}{3^4} \).

Key Concepts

Exponents and Rational Exponents

Exponents represent repeated multiplication of a base number. Rational exponents, such as -4/3, indicate both a root and a power. The numerator indicates the power, while the denominator indicates the root. For example, a negative exponent signifies the reciprocal of the base raised to the positive exponent.

Negative Exponents

A negative exponent indicates that the base should be taken as the reciprocal. For instance, a^(-n) is equivalent to 1/(a^n). This concept is crucial for simplifying expressions with negative exponents, allowing for easier computation and understanding of the expression's value.

Cube Roots

The cube root of a number x, denoted as x^(1/3), is a value that, when multiplied by itself three times, gives x. In the expression 27^(-4/3), the denominator of the rational exponent indicates that we first take the cube root of 27, which is 3, before applying the negative exponent to find the final value.