What is the value of [tex]$\frac{3^m+3^{m+1}}{3^{m+3}-3^{m+1}}$[/tex]?
A) 0
B) [tex]$\frac{1}{6}$[/tex]
C) [tex]$\frac{1}{12}$[/tex]
What is the median of 9, 5, 7, 11, 13, 3?
Answer (1)
Simplify the expression by factoring and canceling common terms: 3 m + 3 − 3 m + 1 3 m + 3 m + 1 = 3 m + 1 ( 3 2 − 1 ) 3 m ( 1 + 3 ) = 3 × 8 4 = 6 1 .
Sort the given numbers in ascending order: 3 , 5 , 7 , 9 , 11 , 13 .
Find the median by averaging the middle two numbers since there are an even number of values: 2 7 + 9 = 8 .
The simplified expression is 6 1 and the median is 8 .
Explanation
Problem Analysis We are given the expression 3 m + 3 − 3 m + 1 3 m + 3 m + 1 and asked to simplify it. We are also asked to find the median of the numbers 9 , 5 , 7 , 11 , 13 , 3 .
Simplifying the Expression Let's simplify the expression first. We can factor out 3 m from the numerator and 3 m + 1 from the denominator: 3 m + 3 − 3 m + 1 3 m + 3 m + 1 = 3 m + 1 ( 3 2 − 1 ) 3 m ( 1 + 3 ) = 3 m + 1 ( 9 − 1 ) 3 m ( 4 ) = 3 m + 1 ( 8 ) 3 m ( 4 ) Now, we can simplify further: 3 m + 1 ( 8 ) 3 m ( 4 ) = 3 ⋅ 8 4 = 24 4 = 6 1 So, the simplified expression is 6 1 .
Finding the Median Now, let's find the median of the numbers 9 , 5 , 7 , 11 , 13 , 3 . First, we need to sort the numbers in ascending order: 3 , 5 , 7 , 9 , 11 , 13 Since there are 6 numbers (an even number), the median is the average of the middle two numbers, which are 7 and 9. So, the median is: 2 7 + 9 = 2 16 = 8
Final Answer Therefore, the value of the expression 3 m + 3 − 3 m + 1 3 m + 3 m + 1 is 6 1 , and the median of the numbers 9 , 5 , 7 , 11 , 13 , 3 is 8 .
Examples
Simplifying expressions and finding medians are fundamental skills in mathematics with applications in various fields. For example, simplifying algebraic expressions is crucial in physics for solving equations of motion, while finding the median is useful in statistics for analyzing data sets, such as determining the central tendency of exam scores or income distributions. These skills provide a foundation for more advanced mathematical concepts and real-world problem-solving.
