Compute: [tex]$\frac{\left(1 \frac{1}{2}\right)^4}{\left(1 \frac{1}{2}\right)^2}$[/tex]
Answer (1)
Convert the mixed number to an improper fraction: 1 2 1 = 2 3 .
Apply the rule for dividing powers with the same base: a n a m = a m − n , resulting in ( 2 3 ) 4 − 2 = ( 2 3 ) 2 .
Compute the square: ( 2 3 ) 2 = 4 9 .
Convert the improper fraction to a mixed number: 4 9 = 2 4 1 . The final answer is 2 4 1 .
Explanation
Understanding the Problem We are asked to compute the value of ( 1 2 1 ) 2 ( 1 2 1 ) 4 . This involves dividing powers with the same base, which simplifies the calculation.
Converting to Improper Fraction First, let's convert the mixed number 1 2 1 to an improper fraction: 1 2 1 = 2 3 . So the expression becomes ( 2 3 ) 2 ( 2 3 ) 4 .
Applying the Power Rule Now, we use the rule for dividing powers with the same base: a n a m = a m − n . In our case, a = 2 3 , m = 4 , and n = 2 . Therefore, ( 2 3 ) 2 ( 2 3 ) 4 = ( 2 3 ) 4 − 2 = ( 2 3 ) 2 .
Squaring the Fraction Next, we compute ( 2 3 ) 2 = 2 2 3 2 = 4 9 .
Converting to Mixed Number Finally, we convert the improper fraction 4 9 back to a mixed number: 4 9 = 2 4 1 .
Final Answer Therefore, the final answer is 2 4 1 .
Examples
Imagine you are scaling a recipe that calls for 1 2 1 cups of flour. If you need to increase the recipe by a factor of ( 1 2 1 ) 2 , you are essentially multiplying the original amount of flour by ( 1 2 1 ) 2 = 2 4 1 . This means you would need 2 4 1 times the original amount of flour. Understanding how to manipulate and simplify expressions with exponents and fractions is crucial in various real-life scenarios, such as cooking, construction, and finance, where scaling quantities is a common task.
