Determine the number that can be used in place of the question mark to make the equation true.
[tex]\left(\frac{1}{2}\right)^?=16[/tex]
A. No solution
B. [tex]$\frac{1}{4}$[/tex]
C. -4
D. 4
Answer (2)
Rewrite 16 as 2 4 .
Express 2 1 as 2 − 1 .
Equate the exponents: − ? = 4 .
Solve for ?: ? = − 4 . The answer is − 4 .
Explanation
Understanding the Problem We are given the equation ( 2 1 ) ? = 16 and asked to find the value of the question mark that makes the equation true. The possible answers are No solution, 4 1 , − 4 , and 4 .
Rewriting the Equation We can rewrite 16 as a power of 2. Since 16 = 2 4 , we can rewrite the equation as ( 2 1 ) ? = 2 4 .
Using Properties of Exponents Since 2 1 = 2 − 1 , we can rewrite the left side of the equation as ( 2 − 1 ) ? = 2 − ? . So the equation becomes 2 − ? = 2 4 .
Solving for the Unknown Now we can equate the exponents: − ? = 4 . Solving for ?, we get ? = − 4 .
Final Answer Therefore, the number that can be used in place of the question mark to make the equation true is − 4 .
Examples
Understanding exponential equations is crucial in various real-world applications, such as calculating the decay of radioactive materials or modeling population growth. For instance, if a radioactive substance has a half-life of 1 year, the amount remaining after t years can be modeled by the equation A ( t ) = A 0 ( 2 1 ) t , where A 0 is the initial amount. Similarly, in finance, compound interest calculations involve exponential growth, where the value of an investment grows exponentially over time.
To solve the equation ( 2 1 ) ? = 16 , we rewrite 16 as 2 4 and 2 1 as 2 − 1 . By equating the exponents, we find that ? = − 4 . Therefore, the correct choice is Option C: -4.
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