Problem 163 Write each number in scientific ... [FREE SOLUTION]

Chapter 5: Problem 163

Write each number in scientific notation. ? 340,000 ? 0.041

Short Answer

Expert verified

340,000 is \( 3.4 \times 10^5 \) and 0.041 is \( 4.1 \times 10^{-2} \).

Step by step solution

Write 340,000 in scientific notation

Identify the significant figures in 340,000, which are 3.4. Count the number of places the decimal point needs to move to get from 3.4 to 340,000. It moves 5 places to the right. Therefore, the scientific notation of 340,000 is written as \[ 3.4 \times 10^5 \]

Write 0.041 in scientific notation

Identify the significant figures in 0.041, which are 4.1. Count the number of places the decimal point needs to move to get from 0.041 to 4.1. It moves 2 places to the right, so it is negative. Therefore, the scientific notation of 0.041 is written as \[ 4.1 \times 10^{-2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Significant Figures

Significant figures, or 'sig figs,' are crucial in scientific notation. They help us understand the precision of a number. The significant figures in a number are all the non-zero digits, plus any zeros between them or after them if there is a decimal point. For example:

In 340,000, the significant figures are 3 and 4, written as 3.4 in scientific notation.
In 0.041, the significant figures are 4 and 1, written as 4.1.

Sig figs exclude leading and trailing zeros without a decimal. Mastering this concept is essential for precision in calculations.

Powers of Ten

Powers of ten are the backbone of scientific notation. They indicate by how much the decimal point has moved. Each movement changes the exponent. Positive powers of ten mean a big number, negative powers mean a small one. For instance:

For 340,000, the decimal point moves 5 places to the right: \[3.4 \times 10^5\]
For 0.041, it moves 2 places to the left (indicating a small number): \[4.1 \times 10^{-2}\]

This system simplifies large and small numbers, making them more manageable and readable.

Decimal Point Movement

Moving the decimal point is key to writing numbers in scientific notation. The trick is to place the decimal after the first significant figure and count the moves:

For 340,000, shift the decimal 5 places right to get \[3.4 \times 10^5\].
For 0.041, shift it 2 places left, resulting in \[4.1 \times 10^{-2}\].

Remember, a right move increases the exponent, while a left move decreases it. With practice, this process becomes second nature!

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Write each number in scientific notation. ? 340,000 ? 0.041

Short Answer

Step by step solution

Write 340,000 in scientific notation

Write 0.041 in scientific notation

Key Concepts

Significant Figures

Powers of Ten

Decimal Point Movement

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