Ch.1 - Introduction: Matter, Energy, and Measurement

Problem 52

Chapter 1, Problem 52

Carry out the following operations and express the answers in exponential notation with the appropriate number of significant figures.
a. 2.791×10⁴ + 8.76×10³
b. 4.67×10²− 5.4437×10⁴
c. (2.481×10⁻² + 7.33×10⁻⁴) × (1.924×10⁻²+ 6.70)
d. (1.3×10⁻⁴− 3.746×10⁻²)/(1.3×10²− 3.746×10⁴)

Verified step by step guidance

a. Align the exponents for the addition: Convert 8.76×10^3 to 0.876×10^4, then add 2.791×10^4 and 0.876×10^4.

a. Perform the addition: 2.791 + 0.876 = 3.667, then express the result in exponential notation as 3.667×10^4.

b. Align the exponents for the subtraction: Convert 4.67×10^2 to 0.0467×10^4, then subtract 5.4437×10^4 from 0.0467×10^4.

b. Perform the subtraction: 0.0467 - 5.4437 = -5.397, then express the result in exponential notation as -5.397×10^4.

c. First, perform the addition inside the parentheses: (2.481×10^−2 + 7.33×10^−4) and (1.924×10^−2 + 6.70), then multiply the results.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Significant Figures

Significant figures are the digits in a number that contribute to its precision. This includes all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal portion. When performing calculations, the result should be reported with the same number of significant figures as the measurement with the least number of significant figures to ensure accuracy.

Exponential Notation

Exponential notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is represented as a product of a number (the coefficient) and a power of ten. For example, 2.791×10^4 means 2.791 multiplied by 10,000. This notation simplifies calculations and helps maintain significant figures.

Order of Operations

The order of operations is a set of rules that dictates the sequence in which calculations should be performed to ensure consistent results. The common acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) helps remember this order. Following these rules is crucial when performing multiple operations, especially in complex expressions involving addition, subtraction, multiplication, and division.

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Textbook Question

Carry out the following operations and express the answers in exponential notation with the appropriate number of significant figures.

a. 2.791×10⁴+ 8.76×10³

b. 4.67×10² − 5.4437×10⁴

c. (2.481×10⁻²+ 7.33×10⁻⁴) × (1.924×10⁻²+ 6.70)

d. (1.3×10⁻⁴ − 3.746×10⁻²)/(1.3×10² − 3.746×10⁴)

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