Rewrite the given equation so there is a single power of 5,000 on each side. Then, set the exponents equal to each other.
[tex]\left(\frac{1}{5,000}\right)^{-2 z} \cdot 5,000^{-2 z+2}=5,000[/tex]
Which equation shows the result?
A. 2=1
B. z=1
C. -4 z+2=1
D. -4 z+1=1
Answer (1)
Rewrite 5 , 000 1 as 5 , 00 0 − 1 .
Simplify the left side using the power of a power rule and the product of powers rule: 5 , 00 0 2 z ⋅ 5 , 00 0 − 2 z + 2 = 5 , 00 0 2 z − 2 z + 2 = 5 , 00 0 2 .
Rewrite the right side as 5 , 00 0 1 .
Equate the exponents: 2 = 1 , so the final answer is 2 = 1 .
Explanation
Understanding the Problem We are given the equation ( 5 , 000 1 ) − 2 z ⋅ 5 , 00 0 − 2 z + 2 = 5 , 000 and we want to rewrite it so there is a single power of 5,000 on each side. Then, we set the exponents equal to each other and identify the resulting equation.
Rewriting the Equation First, we rewrite 5 , 000 1 as 5 , 00 0 − 1 . Substituting this into the equation, we get: ( 5 , 00 0 − 1 ) − 2 z ⋅ 5 , 00 0 − 2 z + 2 = 5 , 000
Applying the Power of a Power Rule Next, we simplify the left side using the power of a power rule, which states that ( a m ) n = a mn . Thus, ( 5 , 00 0 − 1 ) − 2 z = 5 , 00 0 ( − 1 ) ( − 2 z ) = 5 , 00 0 2 z . So the equation becomes: 5 , 00 0 2 z ⋅ 5 , 00 0 − 2 z + 2 = 5 , 000
Applying the Product of Powers Rule Now, we use the product of powers rule, which states that a m ⋅ a n = a m + n . Applying this to the left side, we get: 5 , 00 0 2 z + ( − 2 z + 2 ) = 5 , 000
Simplifying the Exponent Simplifying the exponent on the left side, we have: 5 , 00 0 2 z − 2 z + 2 = 5 , 000 5 , 00 0 2 = 5 , 000
Rewriting the Right Side We can rewrite the right side as 5 , 00 0 1 . So the equation becomes: 5 , 00 0 2 = 5 , 00 0 1
Equating the Exponents Now, we equate the exponents: 2 = 1
Final Answer Therefore, the equation that shows the result is 2 = 1 .
Examples
Understanding exponent rules is crucial in many fields, such as finance and computer science. For instance, when calculating compound interest, the formula involves raising the interest rate to a power representing the number of compounding periods. Similarly, in computer science, understanding powers of 2 is essential for memory allocation and data storage. By mastering these concepts, students can apply them to real-world problems involving exponential growth and decay.
