Verified Answer :

  • The expanded form of the power is \( (-3) \times (-3) \times (-3) \times (-3) \times (-3) \).
  • The numerical value is \(\boxed{-243}\).

Explanation

  1. Understanding the Power Expression

The power \((-3)^5\) involves a base of \(-3\) and an exponent of \(5\). By definition, an exponent \(n\) applied to a base \(b\) means multiplying the base by itself \(n\) times. Therefore, the expanded form of \((-3)^5\) is the product of \(-3\) multiplied by itself five times:
\[(-3) \times (-3) \times (-3) \times (-3) \times (-3)\]

  1. Calculating the Numerical Value

To find the numerical value, we multiply the factors sequentially:

  1. First, compute the product of the first two factors: \((-3) \times (-3) = 9\).
  2. Next, multiply by the third factor: \(9 \times (-3) = -27\).
  3. Then, multiply by the fourth factor: \((-27) \times (-3) = 81\).
  4. Finally, multiply by the fifth factor: \(81 \times (-3) = -243\).

Since the exponent \(5\) is an odd number, the negative sign is preserved in the final product. The value is \(-243\).

Examples

In physics, when dealing with certain scaling laws, you might need to calculate how a quantity changes as you zoom in or out. For example, if a force decreases by a factor of 3 for every unit of distance increased, moving 5 units away results in a factor of \((-3)^5\). Understanding powers helps in predicting these changes and modeling physical phenomena efficiently.

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