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The terms 'base' and 'exponent' are fundamental in understanding powers and exponents. Let's break down what each term means, using the example \((-5)^2\):

- **Base**: The base is the number that is being multiplied. In our example, \(-5\) is the base.

- **Exponent**: The exponent tells us how many times to multiply the base by itself. In \((-5)^2\), the exponent is \2\, meaning we multiply \(-5\) by itself twice.

Putting it all together in the expression \((-5)^2\), we understand it as \(-5 \times -5\). Exponents help simplify and shorten repeated multiplication into a manageable format.

Multiplication is one of the basic operations in mathematics, where we combine equal groups. Here’s a quick overview of multiplication:

1. **Equal Groups**: Think of multiplication as taking equal groups of a certain quantity. For instance, \3 \times 4\ means three groups of four.

2. **Multiplying Bases with Exponents**: When you have an exponent, you are essentially multiplying the base by itself multiple times. As we saw in the problem \((-5)^2\), the exponent \2\ means we multiply \(-5\) by itself: \(-5 \times -5\).

3. **Negative Times Negative**: It’s crucial to know that multiplying two negative numbers yields a positive result. Hence, \(-5 \times -5 = 25\) because:

• Negative times negative equals positive.
• Therefore, \(-5 \times -5 = +25\).

Negative numbers are numbers less than zero. They are shown with a minus sign (\text{{-}}\). Here are some key points to remember about negative numbers, especially in multiplication:

- **Multiplying a Negative by a Positive**: The result is always negative. For example, \-5 \times 3 = -15\.

- **Multiplying Two Negatives**: The result is always positive. This is because the two negative signs cancel each other out, as demonstrated by \(-5)^{2}\ = \(-5 \times -5 = 25\).

- **Using Negatives with Exponents**: When you raise a negative number to an even exponent, the result is positive. But if the exponent is odd, the result is negative. For instance:

• \text{(-2)^2 = 4}\text{ (positive because the exponent is even)}.
• \text{(-2)^3 = -8}\text{ (negative because the exponent is odd)}.

Remember, handling negative numbers correctly in operations is crucial for accurate results.