Rewrite the following numbers as powers:
243
64
Answer (2)
Express 243 as a power of 3: 3 5 = 243 .
Express 64 as a power of 2: 2 6 = 64 .
The number 243 can be written as 3 5 .
The number 64 can be written as 2 6 .
Explanation
Problem Analysis We are asked to rewrite 243 as a power of 3 and 64 as a power of 2. This involves finding the exponents that satisfy the equations 3 x = 243 and 2 y = 64 .
Finding the Power of 3 We need to find the value of x such that 3 x = 243 . Let's test the given options: 3 3 = 27 3 4 = 81 3 5 = 243 So, 3 5 = 243 .
Finding the Power of 2 Next, we need to find the value of y such that 2 y = 64 . Let's test the given options: 2 4 = 16 2 5 = 32 2 6 = 64 So, 2 6 = 64 .
Final Answer Therefore, 243 can be written as 3 5 and 64 can be written as 2 6 .
Examples
Understanding powers is crucial in many areas, such as calculating compound interest. For instance, if you invest 1000 a t anann u a l in t eres t r a t eo f 5 n$ years is given by 1000 ( 1.05 ) n . Here, the exponent n represents the number of years the money is invested. Similarly, powers are used in computer science to measure the complexity of algorithms, where the running time might be proportional to n 2 or 2 n , indicating how the algorithm's performance scales with the input size.
The number 243 can be expressed as 3 5 , and the number 64 can be expressed as 2 6 .
;
