Solve: $\left(\frac{1}{5,000}\right)^{-2 z} \cdot 5,000^{-2 z+2}=5,000$
A. $z=-1$
B. $z=0$
C. $z=1$
D. no solution
Answer (1)
Rewrite the equation using exponent rules.
Simplify the equation to 500 0 2 = 5000 .
Equate the exponents, leading to the contradiction 2 = 1 .
Conclude that there is no solution .
Explanation
Understanding the Problem We are given the equation ( 5 , 000 1 ) − 2 z ⋅ 5 , 00 0 − 2 z + 2 = 5 , 000 and we need to solve for z . We will simplify the equation using exponent rules.
Rewriting the Equation First, we rewrite the term ( 5 , 000 1 ) − 2 z as ( 5000 ) 2 z using the property a − n = a n 1 . The equation becomes ( 5000 ) 2 z ⋅ 5 , 00 0 − 2 z + 2 = 5 , 000 .
Combining Terms Next, we use the property a m ⋅ a n = a m + n to combine the terms on the left side of the equation. This gives us 500 0 2 z + ( − 2 z + 2 ) = 5000 .
Simplifying the Exponent Now, we simplify the exponent on the left side: 2 z + ( − 2 z + 2 ) = 2 z − 2 z + 2 = 2 . So the equation becomes 500 0 2 = 5000 .
Equating Exponents We can rewrite 5000 as 500 0 1 on the right side, so we have 500 0 2 = 500 0 1 . Since the bases are equal, we can equate the exponents: 2 = 1 . This is a contradiction, which means there is no solution to the equation.
Final Answer Therefore, the equation has no solution.
Examples
Imagine you're adjusting the settings on a complex machine where certain parameters need to balance out perfectly. If you find that no matter what you set one dial to, the machine never reaches the desired state, that's similar to having 'no solution' in an equation. This concept is crucial in engineering and system optimization, where finding a solution ensures that all parts of a system work together correctly. Understanding when a system has no solution can save time and resources by preventing futile attempts to find a nonexistent balance.
