Select all the correct answers. Which expressions are equivalent to this exponential expression? [tex]\frac{\left(5^2\right)^{-3} \cdot 5^4}{5^4}[/tex]
[tex]\frac{1}{5}[/tex]
25
[tex]5^2[/tex]
[tex]5^{-1}[/tex]
[tex]5^1[/tex]
Answer (2)
Simplify the given expression using exponent rules: 5 4 ( 5 2 ) − 3 ⋅ 5 4 .
Cancel out 5 4 from the numerator and denominator: ( 5 2 ) − 3 .
Apply the power of a power rule: 5 2 × ( − 3 ) = 5 − 6 .
Check if any of the options are equivalent to 5 − 6 . None of them are, so the final answer is that there are no correct answers among the options.
Explanation
Analyzing the Expression Let's analyze the given expression and simplify it using exponent rules. The expression is: 5 4 ( 5 2 ) − 3 ⋅ 5 4
Simplifying the Numerator First, we simplify the term ( 5 2 ) − 3 . Using the power of a power rule, we get: ( 5 2 ) − 3 = 5 2 × ( − 3 ) = 5 − 6 So the expression becomes: 5 4 5 − 6 ⋅ 5 4
Simplifying the Expression Further Now, we simplify the numerator by multiplying the terms with the same base. Using the rule a m ⋅ a n = a m + n , we have: 5 − 6 ⋅ 5 4 = 5 − 6 + 4 = 5 − 2 So the expression becomes: 5 4 5 − 2
Simplified Expression Next, we simplify the entire expression by dividing terms with the same base. Using the rule a n a m = a m − n , we have: 5 4 5 − 2 = 5 − 2 − 4 = 5 − 6 So the simplified expression is 5 − 6 .
Checking the Options and Identifying Potential Errors Now, let's examine the given options to see which ones are equivalent to 5 − 6 .
5 1 = 5 − 1
25 = 5 2
5 2 = 5 2
5 − 1 = 5 − 1
5 1 = 5 1
None of these options are equal to 5 − 6 . However, we made an error in our simplification. Let's go back to step 2.
We had the expression: 5 4 5 − 6 ⋅ 5 4 Since 5 4 is in both the numerator and the denominator, we can cancel them out. This leaves us with: 5 − 6
So the simplified expression is 5 − 6 .
Let's re-examine the options. None of them are equal to 5 − 6 .
It seems there might be a mistake in the problem or the options provided. However, let's consider a different approach. If we cancel 5 4 from the beginning, we get ( 5 2 ) − 3 = 5 − 6 .
If the original expression was 5 3 ( 5 2 ) − 1 ⋅ 5 4 , then we would have 5 3 5 − 2 ⋅ 5 4 = 5 3 5 2 = 5 − 1 . In this case, 5 − 1 would be a correct answer.
However, given the original expression, none of the options are equivalent to 5 − 6 .
Final Evaluation and Conclusion Upon closer inspection of the original expression, we can cancel out the 5 4 terms in the numerator and denominator directly: 5 4 ( 5 2 ) − 3 ⋅ 5 4 = ( 5 2 ) − 3 = 5 2 ⋅ ( − 3 ) = 5 − 6
Now, let's check if any of the options are equivalent to 5 − 6 :
5 1 = 5 − 1
25 = 5 2
5 2 = 5 2
5 − 1 = 5 − 1
5 1 = 5 1
None of the options match 5 − 6 . Therefore, there are no correct answers among the given options.
Examples
Exponential expressions are used in various fields such as finance, physics, and computer science. For example, calculating compound interest involves exponential growth. If you invest P dollars at an annual interest rate r compounded n times per year, the amount A after t years is given by A = P ( 1 + n r ) n t . Simplifying and understanding exponential expressions helps in predicting investment growth and making informed financial decisions.
The expression 5 4 ( 5 2 ) − 3 ⋅ 5 4 simplifies to 5 − 6 . None of the provided options 5 1 , 25 , 5 2 , 5 − 1 , and 5 1 are equivalent to 5 − 6 . Thus, there are no correct answers among the options listed.
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