Select all the correct answers.
Which expressions are equivalent to this exponential expression?
[tex]$\frac{6^{-10}}{6^{-4}}$[/tex]
[tex]$6^{-6}$[/tex]
[tex]$6^6$[/tex]
[tex]$\frac{1}{6^6}$[/tex]
[tex]$\frac{1}{6^{-6}}$[/tex]
[tex]$\frac{1}{36}$[/tex]
Answer (1)
Simplify the given expression using the exponent rule a n a m = a m − n , resulting in 6 − 6 .
Check if 6 − 6 is an option.
Check if 6 6 1 is equal to 6 − 6 using the property a − n = a n 1 .
The expressions equivalent to 6 − 4 6 − 10 are 6 − 6 and 6 6 1 .
6 − 6 , 6 6 1
Explanation
Understanding the Problem We are given the expression 6 − 4 6 − 10 and asked to find equivalent expressions from the list: 6 − 6 , 6 6 , 6 6 1 , 6 − 6 1 , 36 1 .
Simplifying the Expression To simplify the given expression, we use the rule for dividing exponential expressions with the same base: a n a m = a m − n . Applying this rule, we have 6 − 4 6 − 10 = 6 − 10 − ( − 4 ) = 6 − 10 + 4 = 6 − 6 .
Checking the Options Now we check each of the options to see if they are equal to 6 − 6 .
6 − 6 : This is obviously equal to 6 − 6 .
6 6 : This is not equal to 6 − 6 . In fact, 6 6 = 6 − 6 1 .
6 6 1 : Using the property a − n = a n 1 , we have 6 6 1 = 6 − 6 . So this is equal to 6 − 6 .
6 − 6 1 : Using the property a − n 1 = a n , we have 6 − 6 1 = 6 6 . This is not equal to 6 − 6 .
36 1 : Since 36 = 6 2 , we have 36 1 = 6 2 1 = 6 − 2 . This is not equal to 6 − 6 .
Final Answer Therefore, the expressions equivalent to 6 − 4 6 − 10 are 6 − 6 and 6 6 1 .
Examples
Exponential expressions are used in various fields such as finance, physics, and computer science. For example, in finance, compound interest is calculated using exponential expressions. If you invest a principal amount P 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 . Understanding how to simplify and manipulate exponential expressions is crucial for solving problems related to compound interest and other financial calculations.
