Which expression is equivalent to $\left(\frac{125^2}{125^{\frac{4}{3}}}\right)$?
A. $\frac{1}{25}$
B. $\frac{1}{10}$
C. 10
D. 25
Answer (1)
Use the property a n a m = a m − n to rewrite the expression as 12 5 2 − 3 4 .
Simplify the exponent: 2 − 3 4 = 3 2 , so the expression becomes 12 5 3 2 .
Rewrite 125 as 5 3 , so the expression is ( 5 3 ) 3 2 .
Use the property ( a m ) n = a m × n to simplify to 5 3 × 3 2 = 5 2 = 25 . The final answer is 25 .
Explanation
Understanding the Problem We are given the expression ( 12 5 3 4 12 5 2 ) and asked to find an equivalent expression from the options: 25 1 , 10 1 , 10, 25.
Applying Exponent Rules To simplify the expression, we can use the property of exponents that states a n a m = a m − n . Applying this property, we get: 12 5 3 4 12 5 2 = 12 5 2 − 3 4
Simplifying the Exponent Now, we need to simplify the exponent. We have: 2 − 3 4 = 3 6 − 3 4 = 3 2 So, the expression becomes: 12 5 3 2
Rewriting the Base We can rewrite 125 as 5 3 . Therefore, the expression is: ( 5 3 ) 3 2
Applying Another Exponent Rule Using the property of exponents that states ( a m ) n = a m × n , we can simplify the expression further: ( 5 3 ) 3 2 = 5 3 × 3 2 = 5 2
Calculating the Final Value Finally, we calculate 5 2 :
5 2 = 25 Therefore, the expression ( 12 5 3 4 12 5 2 ) is equivalent to 25.
Examples
Understanding exponents is crucial in many fields, such as finance and computer science. For instance, when calculating compound interest, the formula involves exponents. If you invest 1000 a t anann u a l in t eres t r a t eo f 5 O(n^2)$ means the execution time grows exponentially with the input size n.
