Problem 182 - Introductory Chemistry Atoms First

Question

Write each number in standard notation: (a) \(1.79 \times 10^{-2}\) (b) \(8.76 \times 10^{-9}\) (c) \(4.88 \times 10^{10}\) (d) \(7.52 \times 10^{1}\) (e) \(8.37 \times 10^{\circ}\) (f) \(4.184 \times 10^{4}\)

Step-by-Step Solution

Answer

(a) 0.0179 (b) 0.000\,000\,008\,76 (c) 48\,800\,000\,000 (d) 75.2 (e) 8.37 (f) 41\,840

Step 1: (a) Convert \(1.79 \times 10^{-2}\) to standard notation

To convert a number with a negative exponent to standard notation, divide the coefficient by 10 raised to the power of the absolute value of the exponent. In this case, we divide \(1.79\) by \(10^{2}\). Thus, the number in standard notation is: $$1.79 \times 10^{-2} = \frac{1.79}{10^2} = 0.0179$$

Step 2: (b) Convert \(8.76 \times 10^{-9}\) to standard notation

Similar to step (a), we divide \(8.76\) by \(10^{9}\): $$8.76 \times 10^{-9} = \frac{8.76}{10^9} = 0.000\,000\,008\,76$$

Step 3: (c) Convert \(4.88 \times 10^{10}\) to standard notation

This time the exponent is positive, so we multiply the coefficient by 10 raised to the power of the exponent. Multiply \(4.88\) by \(10^{10}\): $$4.88 \times 10^{10} = 48\,800\,000\,000$$

Step 4: (d) Convert \(7.52 \times 10^{1}\) to standard notation

Multiply the coefficient (\(7.52\)) by 10 raised to the power of the exponent (1): $$7.52 \times 10^{1} = 75.2$$

Step 5: (e) Convert \(8.37 \times 10^{0}\) to standard notation

Any non-zero number raised to the power of 0 is equal to 1. Therefore, multiply \(8.37\) by \(1\): $$8.37 \times 10^0 = 8.37$$

Step 6: (f) Convert \(4.184 \times 10^{4}\) to standard notation

Multiply the coefficient (\(4.184\)) by 10 raised to the power of the exponent (4): $$4.184 \times 10^{4} = 41\,840$$ All numbers are now in standard notation: (a) 0.0179 (b) 0.000\,000\,008\,76 (c) 48\,800\,000\,000 (d) 75.2 (e) 8.37 (f) 41\,840

Key Concepts

Standard Notation

Standard notation is the usual way of writing numbers. It's how we display numbers in everyday life without using any abbreviations or special symbols. When given a number in scientific notation, you convert it to standard notation to represent it in its full form. For example, when you convert a number like \(1.79 \times 10^{-2}\) into standard notation, you end up with a smaller number, expressed in decimals, such as 0.0179. Similarly, larger numbers like \(4.88 \times 10^{10}\) result in very large values when written in standard notation, like 48,800,000,000.

Understanding standard notation is essential as it helps us interpret and communicate numerical data correctly in various contexts, such as scientific papers, financial reports, and everyday transactions.

Exponents

Exponents are a way to express repeated multiplication of a number by itself. They are written as a small number called the exponent above and to the right of a base number. For example, in the expression \(10^5\), 10 is the base and 5 is the exponent, indicating that 10 should be multiplied by itself five times. The result is a large number: 100,000.

Exponents can be applied to any number, not just 10. They simplify expressions and make them easier to read, especially when dealing with very large or very small numbers. In scientific notation, exponents enable us to express numbers compactly, which is extremely useful when handling data in scientific research, mathematics, and engineering.

Positive Exponent

A positive exponent indicates how many times you multiply the base number by itself. In scientific notation, when you see a positive exponent, the number gets larger. For example, \(4.88 \times 10^{10}\) has a positive exponent of 10, meaning you multiply 4.88 by 10 ten times, resulting in the large number 48,800,000,000.

When converting from scientific to standard notation, having a positive exponent essentially shifts the decimal point to the right, as many places as the exponent dictates. This technique is beneficial for dealing with measurements, quantities, or any data where large numbers occur frequently.

Negative Exponent

A negative exponent tells you to divide the base number, typically leading to a smaller result. In scientific notation, when an exponent is negative, it shifts the decimal point to the left. For instance, \(1.79 \times 10^{-2}\) means you divide 1.79 by 10 twice, giving you 0.0179.

Instead of thinking of it as division, you can think of it as moving the decimal point to the left as many places as the exponent suggests. Negative exponents are common when dealing with very tiny quantities, like in fields that require high precision, such as chemistry and physics. Understanding how negative exponents work helps you accurately manipulate and interpret small-scale data.