Question

Given the equation \(w=x y z\), and \(x=1.1 \cdot 10^{3}, y=2.48 \cdot 10^{-2}\), and \(z=6.000\), what is \(w\), in scientific notation and with the correct number of significant figures?

Short answer

Question: Calculate the value of w in scientific notation with the correct number of significant figures, given the equation \(w = xyz\), where \(x = 1.1 \cdot 10^{3}\), \(y = 2.48 \cdot 10^{-2}\), and \(z = 6.000\). Answer: \(w = 1.6 \cdot 10^2\)

Step-by-step solution

Write given numbers in scientific notation

We are given the following values: - \(x = 1.1 \cdot 10^{3}\) - \(y = 2.48 \cdot 10^{-2}\) - \(z = 6.000\)

Multiply \(x\), \(y\), and \(z\) by using the properties of exponents

To multiply these numbers, we can use the following property of exponents: \(a^m \cdot a^n = a^{m+n}\). Using this property: - Multiply the coefficients (\(1.1\), \(2.48\), and \(6\)): \(1.1 \cdot 2.48 \cdot 6 = 16.368\) - Add the exponents of 10: \(3 + (-2) + 0 = 1\)

Combine the results from Step 2

Combining the multiplication of coefficients and the sum of the exponents, we get: \(w = 16.368 \cdot 10^1\)

Convert to proper scientific notation

To write the number in proper scientific notation, the coefficient should be between 1 and 10. So, we need to rewrite \(16.368 \cdot 10^1\) as follows: - Divide \(16.368\) by \(10\) to get \(1.6368\). - Increase the exponent by 1: \(10^1 \cdot 10^1 = 10^2\) Thus, \(w = 1.6368 \cdot 10^2\)

Determine the correct number of significant figures

To find the correct number of significant figures, we have to look back at the given values x, y, and z. We have the following significant figures for these numbers: - \(x\): 2 significant figures (1.1) - \(y\): 3 significant figures (2.48) - \(z\): 4 significant figures (6.000) The rule for multiplication is to use the least number of significant figures in given numbers, which is 2 in this case. So, we need to round \(1.6368\) to 2 significant figures, leaving us with \(w = 1.6 \cdot 10^2\). The final answer is \(w = 1.6 \cdot 10^2\).

Key concepts

Scientific Notation

Understanding scientific notation is crucial when working with very large or very small numbers. It is a method of expressing numbers as a product of two factors: a decimal and a power of ten. For example, the number 650,000 can be written in scientific notation as \(6.5 \times 10^{5}\). This format is particularly useful in science and engineering because it simplifies operations like multiplication and division.

When writing a number in scientific notation, we ensure that there's only one non-zero digit to the left of the decimal point. This is what keeps the number within a 1 to 10 range, thereby making the data manageable and consistent for comparison.

Significant Figures

Significant figures are a way of expressing precision in measured quantities. Every digit in a number is considered significant if it is not merely a placeholder but conveys some precision in measurement. For example, in \(0.00340\), there are three significant figures: 3, 4, and the trailing zero after the decimal point.

To retain precision in calculations, it's important to consider the number of significant figures from the values involved. This ensures that the result of an operation does not imply greater precision than the initial measurements allowed. The general rule is to round the result to the least number of significant figures present in any measurement used in the calculation.

Properties of Exponents

Exponents allow us to represent repeated multiplication concisely. One of the essential properties of exponents, particularly relevant in scientific notation, is the product rule: \(a^{m} \times a^{n} = a^{m+n}\). This property tells us that when we multiply two powers of the same base, we add their exponents.

Another property used in combination with scientific notation is the power of a power rule, which states that \( (a^{m})^{n} = a^{m \times n}\). These properties are not only helpful in simplifying expressions but also critical in maintaining the integrity of scientific number calculations.

Multiplication of Scientific Numbers

When multiplying numbers in scientific notation, we can separate the process into two steps: multiply the decimal coefficients and add the exponents of ten. This utilizes the properties of exponents mentioned earlier and simplifies complex multiplication. In educational terms, this approach breaks down a potentially daunting task into manageable chunks, helping students to better understand and execute the operation.

However, just multiplying correctly isn't enough. We must also consider significant figures to conclude with the proper precision. For instance, if our computed product has more significant figures than any of the original numbers, we must round to match the least precise measurement.