Question

Write each of the following in scientific notation: a. \(180000000 \mathrm{~g}\) b. \(0.00006 \mathrm{~m}\) c. \(750^{\circ} \mathrm{C}\) d. \(0.15 \mathrm{~mL}\) e. \(0.024 \mathrm{~s}\) f. \(1500 \mathrm{~cm}\)

Short answer

a: \(1.8 \times 10^8\), b: \(6 \times 10^{-5}\), c: \(7.5 \times 10^2\), d: \(1.5 \times 10^{-1}\), e: \(2.4 \times 10^{-2}\), f: \(1.5 \times 10^3\).

Step-by-step solution

- Understanding Scientific Notation

Scientific notation is a way to express very large or very small numbers. It has the form: \[ a \times 10^n \] where \(1 ≤ a < 10\) and \(n\) is an integer.

- Converting 180,000,000 g

Move the decimal point in 180,000,000 to the left, until one non-zero digit is before the decimal. This gives: \[ 1.8 \times 10^8 \]

- Converting 0.00006 m

Move the decimal point in 0.00006 to the right, until one non-zero digit is before the decimal. This gives: \[ 6 \times 10^{-5} \]

- Converting 750°C

Move the decimal point in 750 to the left, until one non-zero digit is before the decimal. This gives: \[ 7.5 \times 10^2 \]

- Converting 0.15 mL

Move the decimal point in 0.15 to the right, until one non-zero digit is before the decimal. This gives: \[ 1.5 \times 10^{-1} \]

- Converting 0.024 s

Move the decimal point in 0.024 to the right, until one non-zero digit is before the decimal. This gives: \[ 2.4 \times 10^{-2} \]

- Converting 1500 cm

Move the decimal point in 1500 to the left, until one non-zero digit is before the decimal. This gives: \[ 1.5 \times 10^3 \]

Key concepts

large and small numbers

Understanding how to work with large and small numbers is crucial in many scientific fields. Numbers like 180,000,000 g (grams) and 0.00006 m (meters) can be cumbersome to write and read.
Scientific notation offers a streamlined way to handle these extreme values. This notation simplifies arithmetic operations and helps us grasp the scale of numbers more easily.

decimal point movement

One of the key steps in converting a number to scientific notation is moving the decimal point. Here’s how you do it:

• For large numbers, like 180,000,000 g, you move the decimal to the left until only one non-zero digit remains before it. This gives you: \( 1.8 \times 10^8 \)
• For small numbers, like 0.00006 m, you move the decimal to the right until one non-zero digit is before it. This gives you: \( 6 \times 10^{-5} \)

Moving the decimal point helps in transforming a cumbersome number into a more manageable form.

mathematics in chemistry

In chemistry, scientific notation is especially useful. Chemical quantities can range from incredibly tiny, like the size of an atom, to immensely large, like the number of molecules in a substance. For example:

• A concentration of a solution might be as small as 0.000024 M, which can be written as \( 2.4 \times 10^{-5} \) M.
• The mass of a sample might be 750 g, noted as \( 7.5 \times 10^2 \) g.

Using scientific notation simplifies these diverse measurements, making them easier to manipulate and understand. This is particularly important when dealing with chemical equations and conversions.