Problem 48 - Master Resource Book in JEE Main Physics

Question

If \(3.8 \times 10^{-6}\) is added to \(4.2 \times 10^{-5}\) giving due regard to significant figures, then the result will be [UP SEE 2009] (a) \(4.08 \times 10^{-5}\) (b) \(4.6 \times 10^{-5}\) (c) \(4.5 \times 10^{-5}\) (d) None of these

Step-by-Step Solution

Answer

(b) \(4.6 \times 10^{-5}\)

Step 1: Convert to Standard Form

The numbers given are already in scientific notation. Convert them to standard form for addition. The number \(3.8 \times 10^{-6}\) is equal to \(0.0000038\) and the number \(4.2 \times 10^{-5}\) is equal to \(0.000042\).

Step 2: Align Decimal Places

For easy addition, align the decimal places of the two numbers. Write them such that their decimal points are aligned: \(0.0000038\) and \(0.000042\).

Step 3: Add the Numbers

Add the numbers together:\[ 0.0000038 + 0.000042 = 0.0000458 \]

Step 4: Determine Significant Figures

Identify the number of significant figures in the given numbers. The number \(3.8 \times 10^{-6}\) has two significant figures, and \(4.2 \times 10^{-5}\) has two significant figures, hence the result should be rounded to the smallest number of significant figures, which is two.

Step 5: Round the Sum

Round \(0.0000458\) to two significant figures. The number becomes \(0.000046\).

Step 6: Convert to Scientific Notation

Express the rounded number in scientific notation:\(0.000046\) is equivalent to \(4.6 \times 10^{-5}\).

Key Concepts

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It allows us to handle such numbers conveniently, which is particularly useful in scientific calculations. The basic format is:

• a coefficient (a number greater than or equal to 1 and less than 10),
• multiplied by 10 raised to a power (the exponent).

For example, the number 300,000 can be written in scientific notation as \(3 \times 10^{5}\). Conversely, a small number like 0.00012 would be \(1.2 \times 10^{-4}\). The exponent tells us how many places the decimal has moved:

• a positive exponent means the original number was large,
• a negative exponent indicates a very small original number.

In the given exercise, the numbers \(3.8 \times 10^{-6}\) and \(4.2 \times 10^{-5}\) are both in scientific notation, indicating they are very small numbers due to the negative exponents.

Addition in Scientific Notation

Adding numbers in scientific notation involves a few key steps. To understand this clearly, let's break them down. First, the numbers need to be converted to standard form. This transformation turns them into numbers with fully written-out decimal places, which can be easily aligned for addition. For example,

• \(3.8 \times 10^{-6}\) becomes \(0.0000038\)
• \(4.2 \times 10^{-5}\) becomes \(0.000042\)

Next, aligning the decimal points is crucial. By writing them so their decimal points line up (as shown, \(0.0000038\) with \(0.000042\)), the numbers can be added accurately. After alignment, simply add the numbers together: \[0.0000038 + 0.000042 = 0.0000458\]. Finally, the sum can be converted back into scientific notation if desired, or if specified by the problem.

Rounding Numbers

Rounding numbers is a key skill, especially when dealing with significant figures. It ensures that the results of scientific calculations are as precise as necessary but no more precise than the data warrants. Significant figures are digits that carry meaning contributing to a number's precision. In our example, both original numbers had two significant figures (\(3.8\) and \(4.2\)). Therefore, the final result should also have two significant figures. We round the sum \(0.0000458\) to \(0.000046\). Rounding involves looking at the next digit after the crucial significant figure:- If it's 5 or above, you round up.- If it's less than 5, you leave the last significant digit as it is.Finally, you express this rounded number back in scientific notation: \(0.000046\) becomes \(4.6 \times 10^{-5}\). This keeps the answer concise and precise, reflecting the data's accuracy. Learning to round numbers correctly is invaluable for reporting results that match the precision level indicated by your data.