Evaluate the exponential expression $(-2)^6$.
Answer (1)
The problem asks to evaluate ( − 2 ) 6 .
Since the exponent is even, the result will be positive.
Calculate 2 6 = 2 × 2 × 2 × 2 × 2 × 2 = 64 .
Therefore, ( − 2 ) 6 = 64 .
Explanation
Understanding the problem We are asked to evaluate the exponential expression ( − 2 ) 6 . This means we need to multiply -2 by itself 6 times.
Calculating the value Let's calculate the value: ( − 2 ) 6 = ( − 2 ) × ( − 2 ) × ( − 2 ) × ( − 2 ) × ( − 2 ) × ( − 2 ) Since we have an even number of negative factors, the result will be positive.
Final Calculation Now, let's calculate 2 6 :
2 6 = 2 × 2 × 2 × 2 × 2 × 2 = 64 Therefore, ( − 2 ) 6 = 64 .
Examples
Understanding exponential expressions is crucial in many areas, such as calculating compound interest. For example, if you invest 100 a t anann u a l in t eres t r a t eo f 5 A = P(1 + r)^n , w h ere P$ is the principal amount ( 100 ) , r i s t h e in t eres t r a t e ( 0.05 ) , an d n i s t h e n u mb ero f ye a rs ( 6 ) . T h u s , A = 100(1 + 0.05)^6 = 100(1.05)^6 \approx 134.01$. This shows how exponential growth affects investments over time.
