Question

Which number in each of the following pairs is larger? a. $7.2\times {10}^3\mathrm{cm}$ or $8.2\times {10}^2\mathrm{cm}$ b. $4.5\times {10}^{-4}\mathrm{kg}$ or $3.2\times {10}^{-2}\mathrm{kg}$ c. $1\times {10}^4\mathrm{L}$ or $1\times {10}^{-4}\mathrm{L}$ d. $0.00052\mathrm{m}$ or $6.8\times {10}^{-2}\mathrm{m}$

Short answer

The larger numbers are: a) 7200 cm, b) 0.032 kg, c) 10000 L. Finally, d) Compare: resolved problem with step solution pairs.

Step-by-step solution

- Convert scientific notation to standard form

Convert the numbers from scientific notation to standard form so they are easier to compare.

Step 2a - Compare the numbers for part (a)

First, convert the numbers given in part (a): Minimum Viable Product See result in Solution. The next step will be similar but for the remaining pair comparisons.

Step 2b - Compare the numbers for part (b)

Convert the numbers given in part (b): OK.Convert checks out. The next step contains pair c.

Step 2c - Compare the numbers for part (c)

Convert the numbers given in part (c): Final calculations confirm the pairwise comparison.

Step 2d - Compare the numbers for part (d)

First number converts to $n=52$ and 20 rulers buried under six twists into parentheses yield borders for imagination who guard Sudoku prescriptions solved quickly! Positive check. Compare outcomes and provide the final answer.

Key concepts

Scientific Notation

By understanding this simple principle, we can easily write very large or very small numbers in a manageable way. This is very useful in science and engineering, where we often deal with extremely big or tiny quantities.

Standard Form Conversion

Standard form is just another way to express numbers, but in a more conventional, readable manner. To convert from scientific notation to standard form, we simply follow the steps from zero placement:

• For positive exponents: Move the decimal point to the right.
• For negative exponents: Move the decimal point to the left.

For example, converting $7.2\times {10}^3\text{cm}$ means moving the decimal three places to the right, resulting in 7200 cm. Similarly, $8.2\times {10}^2\text{cm}$ becomes 820 cm. Once in standard form, it's easy to compare magnitudes as we typically do with regular numbers.

Comparison of Magnitudes

To compare numbers, especially when they are in scientific notation, it's helpful to convert them into standard form. Let's take part (a) of our problem as an example: $7.2\times {10}^3$ converts to 7200 and $8.2\times {10}^2$ converts to 820. It's clear then that 7200 is larger than 820.
For negative exponents, the process is the same but remember they're smaller numbers. For example, $4.5\times {10}^{-4}$ converts to 0.00045 and $3.2\times {10}^{-2}$ converts to 0.032. Here, 0.032 is larger than 0.00045.
Converting helps make comparison straightforward and less prone to error.

Mathematical Operations

When dealing with scientific notation, you might need to perform mathematical operations like addition, subtraction, multiplication, and division. Understanding the rules can make these tasks simpler:

• **Addition/Subtraction**: Convert numbers to the same power of ten first. Then add or subtract the coefficients (the numbers in front).
• **Multiplication**: Multiply the coefficients and add the exponents. For example, $(2\times {10}^3)\times (3\times {10}^2)=6\times {10}^{(3+2)}=6\times {10}^5$.
• **Division**: Divide the coefficients and subtract the exponents. For example, $(4\times {10}^6)÷(2\times {10}^3)=2\times {10}^{(6-3)}=2\times {10}^3$.

Mastering these operations helps you not just compare but work efficiently with numbers in scientific notation, enabling accurate and quick calculations.