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In algebra, exponents are used to express repeated multiplication of a number by itself. The expression \( a^n \) represents the base \( a \) being multiplied by itself \( n \) times, where \( n \) is called the exponent. When the base is a fraction, like in the exercise \( \left(-\frac{9}{10}\right)^3 \), the same rule applies. Here, the fraction \( -\frac{9}{10} \) is the base, and the exponent 3 means we multiply \( -\frac{9}{10} \) by itself three times.

It's important to distinguish between \( (-a)^n \) and \( -(a^n) \). In the former, the negative sign is part of the base and is repeated with each multiplication, whereas in the latter, the negative sign is not included in the repeated multiplication and only applies to the final product. Understanding the role of the exponent is crucial as it affects the outcome, especially when dealing with negative bases.

The rules for multiplying negative numbers are straightforward yet vital for understanding expressions like the given exercise. When you multiply two negative numbers, the result is positive, symbolized by the phrase 'a negative times a negative gives a positive'. This is because the negatives cancel each other out. However, multiplying an odd number of negative numbers, as in our exercise with \( -\frac{9}{10} \) multiplied by itself three times, results in a negative number. This is because there's always one unmatched negative to influence the result.

Therefore, when students encounter negative bases with an odd exponent, they can expect a negative result. This knowledge helps untangle complex expressions and is a fundamental concept in algebra that extends to polynomial multiplication and beyond.

Fractional exponents can be intimidating at first glance, but if broken down, they are another way to express roots and powers. A fractional exponent like \( a^{\frac{m}{n}} \) suggests two operations: the nth root of a, followed by raising the result to the mth power, or vice versa. It's equivalent to \( \sqrt[n]{a^m} \) or \( (\sqrt[n]{a})^m \).

When a student encounters a problem with fractional exponents, they should remember this dual operation. This concept is essential when evaluating expressions with fractional exponents, allowing for simplification before multiplying or adding together different terms. It highlights the connection between exponents and roots, offering a different perspective on operations involving powers.