Verified Answer :

  • Identify the equation \(128 = 2^n\) and the need to find the value of \(n\).
  • Recognize that \(128\) is a power of \(2\) by multiplying \(2\) iteratively: \(2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128\).
  • Conclude that \(2^7 = 128\), which implies \(n = 7\).
  • Final answer is \(\boxed{7}\).

Explanation

  1. Understanding the Problem

We are given the equation \(128 = 2^n\), where \(n\) is an unknown exponent. Our goal is to determine the value of \(n\) such that \(2\) raised to the power of \(n\) results in \(128\). We can approach this by expressing \(128\) as a product of the base, which is \(2\).

  1. Expressing 128 as a Power of 2

To express \(128\) as a power of \(2\), we can repeatedly multiply \(2\) by itself and count how many times this occurs:

  • \(2^1 = 2\)
  • \(2^2 = 2 \times 2 = 4\)
  • \(2^3 = 4 \times 2 = 8\)
  • \(2^4 = 8 \times 2 = 16\)
  • \(2^5 = 16 \times 2 = 32\)
  • \(2^6 = 32 \times 2 = 64\)
  • \(2^7 = 64 \times 2 = 128\)

By identifying that \(128\) is the result of multiplying \(2\) by itself \(7\) times, we can write \(128 = 2^7\).

  1. Solving for n

Since \(128 = 2^n\) and we have established that \(128 = 2^7\), we can equate the exponents. Because the base is the same (\(2\)), it must be that \(n = 7\).

Examples

In computer science, data storage is often based on powers of 2 because computers use binary (base-2) arithmetic. For example, a byte consists of 8 bits, which can represent \(2^8 = 256\) different values. Knowing powers of 2 helps in understanding how much data can be stored or addressed in computer memory, making this calculation a foundational skill in digital technology.

Related Questions