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When you encounter an expression with an exponent, it means that you are multiplying the base by itself a certain number of times. For example, in the expression \((-5)^4\), the base is -5, and the exponent is 4. This tells you that -5 should be multiplied by itself four times.

Important points to remember about exponents:

• The base is the number being multiplied.
• The exponent tells you how many times to multiply the base by itself.
• An exponent of 2 is called 'squared,' while an exponent of 3 is called 'cubed.'

In our example, \((-5)^{4}\) means that \(-5\) is multiplied by itself 4 times: \((-5) \times (-5) \times (-5) \times (-5)\).

Multiplying integers can sometimes be tricky, especially when dealing with negative numbers. Here are some rules to keep in mind:

• Multiplying two positive numbers always gives a positive result.
• Multiplying two negative numbers also gives a positive result.
• Multiplying a positive number by a negative number gives a negative result.

Let's apply these rules to our example, \((-5)^4\):

• \((-5) \times (-5) = 25\) (since multiplying two negatives gives a positive)
• \(25 \times (-5) = -125\) (since a positive times a negative gives a negative)
• \(-125 \times (-5) = 625\) (since a negative times a negative gives a positive)

Following these steps, we see that \((-5)^4 = 625\).

Breaking down problems into smaller steps is essential for understanding and solving them correctly. Let’s look at our example \((-5)^{4}\) and see how it is simplified:

Step 1: Understand the Exponent
The expression \(-5^{4}\) tells you to multiply -5 by itself 4 times.
Step 2: Break Down the Multiplication
Write it out as \((-5) \times (-5) \times (-5) \times (-5)\).
Step 3: Multiply Two Factors at a Time
First, multiply the first two factors: \((-5) \times (-5) = 25\)
Next, multiply the result by the next factor: \(25 \times (-5) = -125\)
Finally, multiply by the last factor: \((-125) \times (-5) = 625\)
Step 4: Find the Final Result
Combine all parts to get the final result: \(-5^4 = 625\).

By following these steps, you can simplify even the most complex expressions involving exponents. Remember, practice makes perfect!