Evaluate $(-1)^8+(-1)^7-1^6-1^4-(-1)^2$. Pay attention to the placement of the parentheses and negative signs.
A) -3
B) -23
C) 5
D) -4
Answer (1)
Evaluate each term with exponents: ( − 1 ) 8 = 1 , ( − 1 ) 7 = − 1 , 1 6 = 1 , 1 4 = 1 , ( − 1 ) 2 = 1 .
Substitute the values back into the expression: 1 + ( − 1 ) − 1 − 1 − ( 1 ) .
Simplify the expression: 1 − 1 − 1 − 1 − 1 = − 3 .
The final result is − 3 .
Explanation
Understanding the Expression We are asked to evaluate the expression ( − 1 ) 8 + ( − 1 ) 7 − 1 6 − 1 4 − ( − 1 ) 2 . We need to carefully consider the order of operations and the signs of the numbers.
Evaluating Each Term First, let's evaluate each term individually:
( − 1 ) 8 = 1 because any negative number raised to an even power is positive.
( − 1 ) 7 = − 1 because any negative number raised to an odd power is negative.
1 6 = 1 because 1 raised to any power is 1.
1 4 = 1 because 1 raised to any power is 1.
( − 1 ) 2 = 1 because any negative number raised to an even power is positive.
Substituting the Values Now, substitute these values back into the original expression:
( − 1 ) 8 + ( − 1 ) 7 − 1 6 − 1 4 − ( − 1 ) 2 = 1 + ( − 1 ) − 1 − 1 − ( 1 )
Simplifying the Expression Finally, simplify the expression by performing the addition and subtraction from left to right:
1 + ( − 1 ) − 1 − 1 − ( 1 ) = 1 − 1 − 1 − 1 − 1 = 0 − 1 − 1 − 1 = − 1 − 1 − 1 = − 2 − 1 = − 3
Final Answer Therefore, the value of the expression is -3.
Examples
Understanding how to evaluate expressions with negative numbers and exponents is crucial in many areas of mathematics and physics. For example, when dealing with alternating current (AC) circuits, the voltage and current change direction periodically. The power dissipated in a resistor can be calculated using expressions involving squares of negative numbers, similar to the one in this problem. Correctly evaluating these expressions ensures accurate calculations of power consumption and circuit behavior.
