- Evaluate \((-2)^6 = 64\), \((-8)^2 = 64\), \((-4)^3 = -64\), \(3^4 = 81\), and \(2^6 = 64\).
- Select the expressions that result in 64: \((-2)^6\), \((-8)^2\), and \(2^6\).
- The correct original exponential forms are \(\boxed{(-2)^6, (-8)^2, 2^6}\).
Explanation
- Evaluating Exponential Expressions
To find which of the expressions evaluate to 64, we will evaluate each of the given options individually:
- \((-2)^6\): A negative number raised to an even power results in a positive value. Calculating the expression, we have \((-2) \times (-2) \times (-2) \times (-2) \times (-2) \times (-2) = 4 \times 4 \times 4 = 64\). This is one of the correct forms.
- \((-8)^2\): Similarly, a negative number squared results in a positive value. Calculating this, we get \((-8) \times (-8) = 64\). This is also a correct form.
- \((-4)^3\): A negative number raised to an odd power results in a negative value. Calculating this, we get \((-4) \times (-4) \times (-4) = 16 \times (-4) = -64\). This is not 64.
- \(3^4\): Calculating this, we get \(3 \times 3 \times 3 \times 3 = 9 \times 9 = 81\). This is not 64.
- \(2^6\): Calculating this, we get \(2 \times 2 \times 2 \times 2 \times 2 \times 2 = 64\). This is the final correct form.
- Selecting the Correct Options
Based on our evaluations, the expressions that yield 64 are \((-2)^6\), \((-8)^2\), and \(2^6\). Therefore, these are the three original exponential forms Dawn could have simplified.
Examples
Exponentiation is a fundamental tool used in fields like computer science for calculating memory capacities, growth patterns in finance (like compound interest), and physical phenomena. For example, if a computer's memory chip doubles its capacity every year, the amount of memory after several years is expressed as a power of 2, like \(2^6 = 64\) gigabytes. Understanding these properties helps in quickly estimating and computing values efficiently.