The expression $\left[\left(2^3 \cdot 2^{-5}\right)^3:\left(2^{-4} \cdot 2^2\right)^2\right]^2 \cdot 2^4$ has a value:
A) $1$
B) $2$
C) $1 \text { / } 8$
D) $4$
E) $1 / 2$
Answer (1)
Simplify the expression inside the first parenthesis: 2 3 "." 2 − 5 = 2 − 2 .
Simplify the expression inside the second parenthesis: 2 − 4 "." 2 2 = 2 − 2 .
Simplify the division: 2 − 6 : 2 − 4 = 2 − 2 .
Simplify the expression: 2 − 4 "." 2 4 = 2 0 = 1 .
The final answer is 1 .
Explanation
Understanding the Problem We are asked to find the value of the expression [ ( 2 3 "." 2 − 5 ) 3 : ( 2 − 4 "." 2 2 ) 2 ] 2 "." 2 4 . To do this, we will simplify the expression step by step using the properties of exponents.
Simplifying the First Parenthesis First, we simplify the expression inside the first parenthesis: 2 3 ⋅ 2 − 5 = 2 3 + ( − 5 ) = 2 − 2 .
Simplifying the Second Parenthesis Next, we simplify the expression inside the second parenthesis: 2 − 4 ⋅ 2 2 = 2 − 4 + 2 = 2 − 2 .
Substituting Back into the Original Expression Now we substitute these simplified expressions back into the original expression: [ ( 2 − 2 ) 3 : ( 2 − 2 ) 2 ] 2 ⋅ 2 4 .
Simplifying the Exponents We simplify the exponents: [ 2 − 6 : 2 − 4 ] 2 ⋅ 2 4 .
Simplifying the Division We simplify the division: 2 − 6 : 2 − 4 = 2 − 6 − ( − 4 ) = 2 − 6 + 4 = 2 − 2 .
Substituting Back into the Expression We substitute this simplified division back into the expression: [ 2 − 2 ] 2 ⋅ 2 4 .
Simplifying the Exponent We simplify the exponent: 2 − 4 ⋅ 2 4 .
Simplifying the Expression Finally, we simplify the expression: 2 − 4 ⋅ 2 4 = 2 − 4 + 4 = 2 0 = 1 .
Final Answer Therefore, the value of the expression is 1.
Examples
Understanding exponents is crucial in many fields, such as computer science, where data storage and processing speeds are often expressed in powers of 2. For example, the memory capacity of a computer or the speed of a processor can be described using exponents. Simplifying expressions with exponents helps in optimizing algorithms and understanding the limitations and capabilities of computer systems. This knowledge is also applicable in physics, where exponential growth and decay models are used to describe various phenomena, such as radioactive decay or population growth.
