Julie started to solve the exponential expression $(-2)^3$. $(-2)^3=-2 \cdot-2 \cdot-2=?$
Which of these explains why Julie gets a negative number for her final answer?
A. The base -2 is a negative number.
B. The exponent 3 is a positive number.
C. The exponent 3 is an odd number.
D. The base -2 is an even number.
Answer (1)
Calculate ( − 2 ) 3 = − 8 .
Analyze why the result is negative based on the given options.
Determine that a negative number raised to an odd power is negative.
Conclude that the exponent 3 being an odd number explains the negative result: The exponent 3 is an odd number.
Explanation
Calculate the expression We are asked to determine why the expression ( − 2 ) 3 results in a negative number. Let's first calculate the value of the expression: ( − 2 ) 3 = − 2 × − 2 × − 2 = 4 × − 2 = − 8 . So, the result is indeed a negative number.
Analyze the options Now, let's analyze the given options:
The base -2 is a negative number: While true, this doesn't fully explain why the result is negative. A negative base can result in a positive or negative number depending on the exponent.
The exponent 3 is a positive number: This is also true, but doesn't determine the sign of the result. Positive exponents can result in positive or negative numbers depending on the base.
The exponent 3 is an odd number: This is the correct explanation. When a negative number is raised to an odd power, the result is always negative. This is because each pair of negative numbers multiplied together results in a positive number, and with an odd exponent, there will always be one negative number left over.
The base -2 is an even number: This is not relevant. While -2 is an even integer, the term 'even number' doesn't explain why the result is negative. The key factor is whether the exponent is odd or even.
Conclusion Therefore, the reason Julie gets a negative number for her final answer is that the exponent 3 is an odd number.
Examples
Understanding why negative numbers raised to odd powers result in negative numbers is crucial in various fields. For example, in physics, when dealing with alternating current (AC) circuits, the power dissipated can sometimes be represented using exponential functions with negative bases. If the exponent represents time and is an odd integer, knowing the result will be negative helps in interpreting whether energy is being consumed or generated at that specific time. This concept also applies in financial modeling, where negative growth rates over odd periods can indicate losses in investments.
