Question

Evaluate each of the following mathematical expressions, and express the answer to the correct number of significant digits. a. \((4.771+2.3) / 3.1\) b. \(5.02 \times 10^{2}+4.1 \times 10^{2}\) c. \(1.091 \times 10^{3}+2.21 \times 10^{2}+1.14 \times 10^{1}\) d. \(\left(2.7991 \times 10^{-6}\right) /\left(4.22 \times 10^{6}\right)\)

Short answer

Answer: a. 2.3 b. 910 c. \(1.3234 \times 10^{3}\) d. \(0.663 \times 10^{-12}\)

Step-by-step solution

a. \((4.771+2.3) / 3.1\)

First, we will add 4.771 and 2.3 while keeping significant digits in mind. Since 2.3 has fewer decimal places (1) compared to 4.771 (3), the sum should have only 1 decimal place. Then, we will divide the result by 3.1 and ensure that the quotient has the correct number of significant digits. 1. Add 4.771 and 2.3: \(4.771 + 2.3 = 7.1\) 2. Divide the sum by 3.1: \(7.1 ÷ 3.1 ≈ 2.3\) Answer: \(2.3\)

b. \(5.02 \times 10^{2}+4.1 \times 10^{2}\)

Since both terms have the same power of 10, we can add the coefficients (5.02 and 4.1) and keep the power of 10 intact. The sum of the coefficients should have the same number of decimal places as the term with the least decimal places (1). 1. Add the coefficients: \(5.02 + 4.1 = 9.1\) 2. Multiply with the power of 10: \(9.1 \times 10^{2} = 910\) Answer: \(910\)

c. \(1.091 \times 10^{3}+2.21 \times 10^{2}+1.14 \times 10^{1}\)

First, we will convert each term to the same power of 10, the highest one in this case (10^3). Then, we will add the coefficients, keeping the right number of significant digits. Finally, we will express the answer in the correct scientific notation form. 1. Convert to same power of 10: - \(2.21 \times 10^{2} = 0.221 \times 10^{3}\) - \(1.14 \times 10^{1} = 0.0114 \times 10^{3}\) 2. Add the coefficients: - \(1.091 + 0.221 + 0.0114 = 1.3234\) 3. Convert to scientific notation (no need in this case since it is already in the correct form): \(1.3234 \times 10^{3}\) Answer: \(1.3234 \times 10^{3}\)

d. \(\left(2.7991 \times 10^{-6}\right) /\left(4.22 \times 10^{6}\right)\)

To divide two numbers in scientific notation, we can divide their coefficients and subtract their exponents. 1. Divide the coefficients: \(\frac{2.7991}{4.22} ≈ 0.663\) 2. Subtract the exponents: \((-6) - (6) = -12\) 3. Combine the result: \(0.663 \times 10^{-12}\) Answer: \(0.663 \times 10^{-12}\)

Key concepts

Understanding Scientific Notation

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It's particularly useful in fields like science and engineering where such numbers are common. Scientific notation has two parts: a coefficient and a power of ten. For example, in the expression \(5.02 \times 10^{2}\), \(5.02\) is the coefficient and \(10^{2}\) indicates that the coefficient should be multiplied by 100 (or \(10\) raised to the power of 2).

When performing calculations, it's important to remember that you can only add or subtract numbers in scientific notation if they have the same exponent. In the case of multiplication and division, you can manipulate the exponents accordingly, multiplying adds the exponents while dividing subtracts them. For the correct significant digits, the final answer should reflect the same precision as the least precise number you're working with.

Evaluating Mathematical Expressions

Evaluating mathematical expressions involves performing operations like addition, subtraction, multiplication, and division in the correct order, following the rules of arithmetic and algebra. This is crucial in scientific notation where the significance of digits plays a big role. When combining numbers, pay special attention to significant digits which are all the non-zero digits plus any trailing or sandwiched zeros, as they indicate precision.

For example, when adding \(4.771\) and \(2.3\), we must round to the least number of decimal places; here it's one decimal place, leading to \(7.1\). In scientific notation, when adding or subtracting, the terms must be converted to the same power of ten before combining them. For instance, converting \(2.21 \times 10^{2}\) to \(0.221 \times 10^{3}\) aligns it with \(1.091 \times 10^{3}\) for addition.

Chemistry Calculations and Significant Digits

In chemistry calculations, significant digits are essential because they convey the precision of measurements. Chemists must report their answers with the right number of significant digits to accurately reflect the certainty of their experimental data. For instance, the digit in a measurement like \(2.3\) mL, which has two significant digits, suggests a degree of measurement accuracy to the nearest tenth of a milliliter.

During calculations, it’s important to consider the least number of significant digits in any number involved. For example, when dividing \(2.7991 \times 10^{-6}\) by \(4.22 \times 10^{6}\), the answer is reported with the same number of significant digits as the least precise measurement, in this case, three significant digits (from 4.22), resulting in \(0.663 \times 10^{-12}\). This concept ensures that the reported result does not overstate its accuracy.