Question

Express the following in scientific notation: (a) \(132.5 \mathrm{~cm}\) (b) \(0.000883 \mathrm{~km}\) (c) \(6,432,000,000 \mathrm{nm}\)

Short answer

Question: Express the following numbers in scientific notation: (a) 132.5 cm (b) 0.000883 km (c) 6,432,000,000 nm Answer: (a) \(1.325 \times 10^1 \:\mathrm{cm}\) (b) \(8.83 \times 10^{-4} \:\mathrm{km}\) (c) \(6.432 \times 10^9 \:\mathrm{nm}\)

Step-by-step solution

Find single digit

First, we need to move the decimal point to make a number between 1 and 10. In this case, we move the decimal one place to the left: \(1.325\).

Determine the power of 10

Since we moved the decimal point one place to the left, the power of 10 will be 1.

Write the final expression

Now, we write the number in scientific notation: \(1.325 \times 10^1 \mathrm{~cm}\). #problem(b)# (b) \(0.000883 \mathrm{~km}\)

Find single digit

Move the decimal point to the right to make a number between 1 and 10: \(8.83\).

Determine the power of 10

Since we moved the decimal point 4 places to the right, the power of 10 will be -4.

Write the final expression

We write the number in scientific notation: \(8.83 \times 10^{-4} \mathrm{~km}\). #problem(c)# (c) \(6,432,000,000 \mathrm{nm}\)

Find single digit

Move the decimal point to the left to make a number between 1 and 10: \(6.432\).

Determine the power of 10

Since we moved the decimal point 9 places to the left, the power of 10 will be 9.

Write the final expression

We write the number in scientific notation: \(6.432 \times 10^9 \mathrm{~nm}\).

Key concepts

Expressing Numbers in Scientific Notation

Scientific notation is a convenient way to express very large or very small numbers. It simplifies numbers by using powers of 10. To convert a number to scientific notation, you start by creating a new number that’s between 1 and 10 by moving the decimal point.

For example, for the number 132.5, we moved the decimal to the left to get 1.325. This is because our new number should remain as close as possible to the original in magnitude but should fall between 1 and 10. The number of places you moved the decimal point determines the exponent of the 10. Since we moved it one place, the exponent is 1, and we express 132.5 cm as \(1.325 \times 10^1 \) cm in scientific notation.

Powers of 10

When expressing numbers in scientific notation, we use powers of 10. The power, or exponent, can be positive or negative depending on whether the number is large or small. A positive exponent shows how many times we multiply 10 to get the number, while a negative exponent indicates how many times we divide 10 to reach the number.

About the exercise example (b), 0.000883 km is written as \(8.83 \times 10^{-4} \) km. The exponent -4 shows that we moved the decimal point 4 places to the right, which is the same as dividing by 10,000. Knowing how to manipulate the decimal point and understanding the meaning of positive and negative exponents is key in mastering scientific notation.

Significant Figures

Significant figures are important in scientific notation because they convey the precision of a measurement. Each non-zero digit in a reading is considered significant. Zeros can also be significant depending on their position. When converting to scientific notation, it’s important to retain all significant figures from the original number.

In problem (c), we convert 6,432,000,000 nm to scientific notation as \(6.432 \times 10^9 \) nm. The zeros after 6432 are not significant; they are just placeholding zeros, which do not affect the precision of the original number. In scientific notation, we drop these zeros, but the power of 10 (in this case, 9) captures the magnitude of the number correctly. Remember, while zeros within a number or those trailing non-zero digits in a number with a decimal point are significant, leading zeros and trailing zeros in a large number without a decimal point are not.