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Negative exponents can initially seem confusing, but they're quite simple when you understand the core concept. A negative exponent indicates that you take the reciprocal of the base raised to the positive exponent.
For example, let's consider the term \(5^{-2}\). Instead of multiplying 5 by itself, you convert it to \(\frac{1}{5^2}\). This means that \(5^{-2} = \frac{1}{25}\). By understanding this, you can simplify expressions with negative exponents straightforwardly.
In the given exercise, the step-by-step solution simplifies: 3 \(\cdot 5^{-2} = 3 \cdot \frac{1}{25} = \frac{3}{25}\). The negative exponent on 5 changed the calculation from multiplication involving large numbers to manageable fractions.

Multiplication is one of the fundamental operations in prealgebra. It combines numbers to get a product. In our initial problem \(3 \cdot 5^{-2}\), we see multiplication with both a constant and an expression that simplifies to a fraction.
To multiply, you follow standard arithmetic rules:

• Multiply the numerators of the fractions.
• Multiply the denominators of the fractions.

In the example, after applying the rules for negative exponents, the multiplication is 3 times \(\frac{1}{25}\), which results in \(\frac{3}{25}\). This multiplication step is straightforward once the expression with a negative exponent is simplified.

Understanding the order of operations is crucial when simplifying expressions, especially with multiple operations.
The mnemonic PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) can help you remember the sequence. Following PEMDAS ensures that you simplify expressions correctly.
Let's take a look at the second part of the exercise ((3 \cdot 5)^{-2}). According to PEMDAS, we must first simplify inside the parentheses:

• Multiply 3 and 5 to get 15.

Next, apply the negative exponent to the result:

• Calculate \(15^{-2} = \frac{1}{15^2} = \frac{1}{225}\).

By adhering to the order of operations, complex expressions become simpler and more manageable.