Write $2^{20}$ as an exponent with the base of $2^5$.
Answer (1)
Apply the power of a power rule: ( a m ) n = a mn , so ( 2 5 ) x = 2 5 x .
Equate the exponents: 5 x = 20 .
Solve for x : x = 5 20 = 4 .
The expression 2 20 can be written as ( 2 5 ) 4 , so the exponent is 4 .
Explanation
Understanding the Problem We are given the expression 2 20 and we want to rewrite it in the form ( 2 5 ) x , where x is the exponent we need to find.
Applying the Power of a Power Rule Using the power of a power rule, which states that ( a m ) n = a mn , we can rewrite ( 2 5 ) x as 2 5 x . So, we have the equation 2 5 x = 2 20 .
Equating the Exponents Since the bases are equal, we can equate the exponents: 5 x = 20 .
Solving for x Now, we solve for x by dividing both sides of the equation by 5: x = 5 20 = 4
Final Answer Therefore, 2 20 can be written as ( 2 5 ) 4 .
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with memory sizes (e.g., kilobytes, megabytes, gigabytes) or in finance when calculating compound interest. For instance, if you invest $100 at an annual interest rate of 5% compounded quarterly, the amount after 5 years can be calculated using exponents. Similarly, in computer science, the number of bits needed to represent a certain amount of data grows exponentially, making the understanding of exponents essential for efficient data storage and processing.
