Verified Answer :

  • Identify the first non-zero digit in \(0.0000115741\), which is 1.
  • Shift the decimal point 5 places to the right to place it after the first non-zero digit, obtaining the significand \(1.15741\).
  • Assign the exponent \(-5\) because the decimal was shifted 5 places to the right.
  • Express the number as the product of the significand and the power of 10: \(\boxed{1.15741 \times 10^{-5}}\).

Explanation

  1. Analysis of the Problem

The number to be expressed in scientific notation is \(0.0000115741\). Scientific notation represents a number as a product of a number (the significand) between 1 (inclusive) and 10 (exclusive) and an integer power of 10. The goal is to write this as \(a \times 10^n\), where \(1 \le |a| < 10\) and \(n\) is an integer.

  1. Converting to Scientific Notation

To convert \(0.0000115741\) to the form \(a \times 10^n\), we need to shift the decimal point to the right until it is immediately after the first non-zero digit, which is '1'. Counting the moves:

  • Moving 1 place to the right gives \(0.000115741\).
  • Moving 2 places to the right gives \(0.00115741\).
  • Moving 3 places to the right gives \(0.0115741\).
  • Moving 4 places to the right gives \(0.115741\).
  • Moving 5 places to the right gives \(1.15741\).

The significand is \(1.15741\). Since we moved the decimal point 5 places to the right, the exponent of 10 is \(-5\).

  1. Final Result

Combining the significand and the power of 10, the number \(0.0000115741\) in scientific notation is \(1.15741 \times 10^{-5}\). Thus, the final answer is \(\boxed{1.15741 \times 10^{-5}}\).

Examples

Scientific notation is widely used in science and engineering to handle extremely large or small numbers. For instance, in biology, scientists might express the length of a bacterium in meters as \(1.5 \times 10^{-6} \text{ m}\) or the mass of a molecule as \(3.3 \times 10^{-23} \text{ grams}\). By using scientific notation, researchers avoid writing strings of zeros, which makes calculations less prone to errors and numbers much easier to compare.

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