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When you encounter a negative exponent in an expression, it can seem a bit puzzling at first. Negative exponents actually tell you to take the reciprocal of the base number. In simpler terms, when you have something like \( a^{-n} \), this is the same as \( \frac{1}{a^n} \). This transformation makes calculations a lot easier.For example, let's say you have \( (-\frac{1}{27})^{-5/3} \). The negative exponent instructs us to take the reciprocal of \(-\frac{1}{27}\), which becomes \(-27\). Now, the exponent is positive, making it further calculation-friendly.

Fractional exponents might look complicated, but they offer a great way to express roots conveniently. A fractional exponent like \( a^{m/n} \) means you first take the n-th root of \( a \) and then raise it to the power of \( m \).

• \( a^{1/n} \) represents the n-th root of \( a \).
• \( a^{m/n} \) represents \( (a^{1/n})^m \).

Let's apply this idea to our exercise. Once we simplified our base with the negative exponent, we had \((-27)^{5/3}\). Here, \(5/3\) means we first find the cube root (since 3 is in the denominator) and then raise the result to the power of 5 (since 5 is in the numerator). So, \((-27)^{1/3} = -3\), and then we raise \(-3\) to the power of \( 5 \), as shown in the simplified solution steps.

Understanding cube roots is key when working with fractional exponents. The cube root of a number \( x \) is the value that, when multiplied by itself three times, returns \( x \). Denoted as \( x^{1/3} \) in mathematical terms, it simplifies expressions directly tied to cubic calculations.For this reason, when you're working with \((-27)^{5/3}\), finding the cube root of \(-27\) becomes the first step. Therefore, the cube root of \(-27\) is \(-3\), since \(-3 \times -3 \times -3 = -27\). Once you've identified the cube root, the next step is to utilize it to further power calculations as required, such as raising it to the fifth power to complete the process in the given example.