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Chapter 5: Problem 52

\(\left(1.5 \times 10^{4} \mathrm{~cm}\right)\left(4 \times 10^{7} \mathrm{~cm}\right)\)

Short Answer

Expert verified
The result is \(6 \times 10^{11} \text{ cm}^2\).

Step by step solution

01

Identify the Numbers and Units

The problem involves multiplying two numbers: \(1.5 \times 10^4 \text{ cm}\) and \(4 \times 10^7 \text{ cm}\). Both numbers are expressed in scientific notation and have the same unit: centimeters (cm).
02

Multiply the Coefficients

Multiply the coefficients (the numbers in front of the powers of 10). In this case: \(1.5 \times 4 = 6\).
03

Add the Exponents

Add the exponents of the powers of 10. Here, \(10^4\) and \(10^7\). So, \(10^4 \times 10^7 = 10^{4+7} = 10^{11}\).
04

Combine the Results

Combine the result of the coefficient multiplication and the power of 10. This results in: \(6 \times 10^{11} \text{ cm}^2\).

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Key Concepts

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

Scientific Notation
Scientific notation is a method of writing very large or very small numbers in a concise form. It consists of two parts: a coefficient and a power of 10. The coefficient is a number greater than or equal to 1 and less than 10. The power of 10 indicates how many times the number must be multiplied or divided by 10.
For example, the number 1,500 can be written as \(1.5 \times 10^3\). This format simplifies calculations, especially with large-scale numbers or very precise measurements. Scientists, engineers, and mathematicians often use scientific notation to handle data more efficiently.
Exponent Addition
When multiplying numbers in scientific notation, you need to add the exponents of the powers of 10. This process comes from one of the basic exponent rules: when you multiply two exponential terms with the same base, you add the exponents.
For example, if you have \(10^4\) and \(10^7\), multiplying them gives you \(10^{4+7}\). So, \(10^4 \times 10^7 = 10^{11}\). This rule helps to keep the computation simple and straightforward, letting you focus on multiplying the coefficients.
Coefficient Multiplication
Coefficient multiplication in scientific notation involves multiplying the numbers in front of the powers of 10. These numbers are called coefficients, and they must be managed separately from the exponents.
In the given problem, the coefficients are 1.5 and 4. To multiply these coefficients, you simply find their product: \(1.5 \times 4 = 6\). This step ensures that your final number retains the correct scale before you add the exponents.
Unit Conversion
Units play a crucial role in scientific notation, especially in fields like physics and engineering. In problems involving multiplication, it's essential that the units are consistent. In our problem, the units are centimeters (cm). When multiplying, units also get squared, just like the numbers.
Upon multiplying \(1.5 \times 10^4 \text{ cm}\) and \(4 \times 10^7 \text{ cm}\), you end up with \(6 \times 10^{11} \text{ cm}^2\). This indicates the area, showing how the dimensions (in this case length) multiply to form a two-dimensional measure (area).

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