Simplify each expression completely.
a) $-2^2$
Answer
b) $(-4)^4$
Answer
c) $-(-4)^3$
Answer (2)
− 2 2 evaluates to − ( 2 2 ) = − 4 .
( − 4 ) 4 evaluates to 256 because a negative number to an even power is positive.
− ( − 4 ) 3 evaluates to − ( − 64 ) = 64 because a negative number to an odd power is negative, and then we take the negative of that.
The simplified expressions are − 4 , 256 , and 64 .
Explanation
Understanding the Problem We will simplify each expression by applying the order of operations, paying close attention to the placement of negative signs and exponents.
Simplifying a) Expression a) − 2 2 means − ( 2 2 ) . The exponent 2 applies only to the number 2, not the negative sign. So, we calculate 2 2 = 2 × 2 = 4 , and then apply the negative sign to get − 4 .
Simplifying b) Expression b) ( − 4 ) 4 means ( − 4 ) × ( − 4 ) × ( − 4 ) × ( − 4 ) . Since we have an even exponent, the result will be positive. ( − 4 ) × ( − 4 ) = 16 , and 16 × 16 = 256 . So, ( − 4 ) 4 = 256 .
Simplifying c) Expression c) − ( − 4 ) 3 means − (( − 4 ) 3 ) . First, we calculate ( − 4 ) 3 = ( − 4 ) × ( − 4 ) × ( − 4 ) = − 64 . Then, we apply the negative sign outside the parentheses: − ( − 64 ) = 64 . So, − ( − 4 ) 3 = 64 .
Final Answer Therefore, the simplified expressions are: a) − 2 2 = − 4 b) ( − 4 ) 4 = 256 c) − ( − 4 ) 3 = 64
Examples
Understanding how exponents and negative signs interact is crucial in many areas, such as physics and engineering. For example, when calculating the potential energy of a system, you might encounter terms like − x 2 , where it's important to correctly evaluate the square before applying the negative sign. Similarly, in electrical engineering, calculating power involves squaring current or voltage, and correctly handling negative values is essential for accurate results. These basic algebraic manipulations are fundamental to more complex calculations in various scientific and technical fields.
The simplified expressions are -4 for − 2 2 , 256 for ( − 4 ) 4 , and 64 for $-(-4)^3. Understanding the application of negative signs and exponents is crucial in these calculations.
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