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)
Apply the quotient of powers property: 12 5 3 4 12 5 2 = 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 : ( 5 3 ) 3 2 .
Apply the power of a power property and calculate: ( 5 3 ) 3 2 = 5 3 ⋅ 3 2 = 5 2 = 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 the Quotient of Powers Property We will use the properties of exponents to simplify the given expression. The key property we'll use is the quotient of powers property, which states that a n a m = a m − n .
Simplifying the Exponent Applying the quotient of powers property to the given expression, we have: 12 5 3 4 12 5 2 = 12 5 2 − 3 4 .
Rewriting 125 Now, we simplify the exponent: 2 − 3 4 = 3 6 − 3 4 = 3 2 .So the expression becomes 12 5 3 2 .
Applying the Power of a Power Property We can rewrite 125 as 5 3 , so the expression is ( 5 3 ) 3 2 .
Calculating the Final Value Now we use the power of a power property, which states that ( a m ) n = a mn . Applying this property, we get: ( 5 3 ) 3 2 = 5 3 ⋅ 3 2 = 5 2 .
Conclusion Finally, we calculate 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. For example, calculating compound interest involves using exponents to determine the future value of an investment. If you invest 100 a t anann u a l in t eres t r a t eo f 5 A = P(1 + \frac{r}{n})^{nt} , w h ere A i s t h e f u t u re v a l u e , P i s t h e p r in c i p a l am o u n t , r i s t h e ann u a l in t eres t r a t e , ni s t h e n u mb ero f t im es t h e in t eres t i sco m p o u n d e d p erye a r , an d t i s t h e n u mb ero f ye a rs . I n t hi sc a se , A = 100(1 + \frac{0.05}{4})^{4 \cdot 10} \approx 164.36$. Exponents are also used in calculating growth rates, depreciation, and other financial metrics.
