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

Step-by-step solution

Identify the number of significant figures in the given values

Each of the given values is in scientific notation. To find the number of significant figures in each value, we look at the non-zero digits. For \(x = 1.1 \cdot 10^3,\) there are 2 significant figures. For \(y = 2.48 \cdot 10^{-2},\) there are 3 significant figures, and for \(z = 6.000,\) there are 4 significant figures.

Calculate the value of \(w\) without worrying about significant figures

We just need to plug the values of \(x,\) \(y,\) and \(z\) into the equation and perform the multiplication: \(w = 1.1 \cdot 10^3 \times 2.48 \cdot 10^{-2} \times 6.000\) First, we can multiply the coefficients and then multiply the powers of 10: \(w = (1.1 \times 2.48 \times 6.000) \times (10^3 \times 10^{-2} \times 1)\) \(w = 16.368 \times 10^1\)

Convert the calculated value of \(w\) to scientific notation

The calculated value of \(w\) needs to be put in scientific notation before we can determine the right number of significant figures: \(w = 1.6368 \cdot 10^2\)

Consider the least number of significant figures

In a multiplication, the number of significant figures in the result should be equal to the least number of significant figures among the given values. In this case, we have 2, 3, and 4 significant figures for \(x,\) \(y,\) and \(z,\) respectively. The least number of significant figures is 2.

Round the value of \(w\) to the correct number of significant figures

We need to round the value of \(w\) to have 2 significant figures: \(w = 1.6 \cdot 10^2\)

Write the final answer

The value of \(w\) in scientific notation and with the correct number of significant figures is: \(w = 1.6 \cdot 10^2\)

Key concepts

Scientific Notation

Understanding scientific notation is crucial in fields ranging from physics to engineering, where it allows us to express very large or very small numbers succinctly. Scientific notation takes the form of \( M \times 10^n \), where \( M \) is a number greater than or equal to 1 and less than 10, and \( n \) is an integer. This format is invaluable when dealing with the scale of measurements required, presenting them in a format easy to read and compute.

For instance, in the given exercise, \( x = 1.1 \cdot 10^3 \) illustrates scientific notation with \( 1.1 \) being the coefficient and \( 3 \) the exponent of ten, dictating the value's magnitude. Similarly, \( y = 2.48 \cdot 10^{-2} \) and \( z = 6.000 \) are presented in a manner where the exponent of ten indicates how the decimal point moves, signifying the actual size of the number.

Significant Figures Calculation

Significant figures are used to indicate the precision of a measured or calculated quantity. In the calculation of \( w \) from the provided values of \( x \), \( y \) and \( z \) in our initial equation, we first determine how many significant figures are present in each number. The non-zero digits are typically considered significant. For example, in \( x = 1.1 \cdot 10^3 \) there are two significant figures. Understanding that zeroes can also be significant depending on their position is key—leading zeroes are not significant, whereas trailing zeroes in a decimal number are. Therefore, \( z = 6.000 \) has four significant figures.

Why are significant figures important?

• They reflect the accuracy of a measurement or calculation.
• They guide the rounding of numbers to avoid overrepresentation of precision.
• Determining the correct number of significant figures impacts the final results in calculations, particularly in a scientific or engineering context.

Multiplication with Significant Figures

When multiplying numbers as in our exercise, the rule for significant figures is simple but strict: the final answer must have the same number of significant figures as the quantity with the least significant figures among those being multiplied. This prevents overstating the precision of the result because the accuracy of a calculation is only as good as the least accurate number used in that calculation.

For instance, when we calculate \( w = 1.1 \cdot 10^3 \times 2.48 \cdot 10^{-2} \times 6.000 \) our solution is initially \( 16.368 \times 10^1 \) which is then converted to scientific notation as \( 1.6368 \cdot 10^2 \). Here, \( x \) dictates the precision with its two significant figures, leading us to round our result to \( 1.6 \cdot 10^2 \) to maintain consistent accuracy. This application of significant figures in multiplication is vital in ensuring that the reliability of our calculations reflects the limits of the given data.