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.