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Chapter 2: Problem 13

Solve each problem, and express the answer in scientific notation. a. \(\left(5 \times 10^{-5}\right)+\left(2 \times 10^{-5}\right) \quad\) c. \(\left(9 \times 10^{2}\right)-\left(7 \times 10^{2}\right)\) b. \(\left(7 \times 10^{8}\right)-\left(4 \times 10^{8}\right) \quad\) d. \(\left(4 \times 10^{-12}\right)+\left(1 \times 10^{-12}\right)\)

Short Answer

Expert verified
a. \(7 \times 10^{-5}\); b. \(3 \times 10^{8}\); c. \(2 \times 10^{2}\); d. \(5 \times 10^{-12}\).

Step by step solution

01

Align Terms with Same Exponent for Problem a

To solve the problem \( \left(5 \times 10^{-5}\right)+\left(2 \times 10^{-5}\right) \), notice that both terms already have the same base and exponent. So, we can simply add the coefficients: \(5 + 2 = 7\).
02

Write Result in Scientific Notation for Problem a

With the added coefficients, rewrite the result in scientific notation: \(7 \times 10^{-5}\).
03

Align Terms with Same Exponent for Problem c

Consider the problem \( \left(9 \times 10^{2}\right)-\left(7 \times 10^{2}\right) \). Both terms have the same base and exponent, so subtract the coefficients: \(9 - 7 = 2\).
04

Write Result in Scientific Notation for Problem c

Write the subtracted result in scientific notation: \(2 \times 10^{2}\).
05

Align Terms with Same Exponent for Problem b

For the problem \( \left(7 \times 10^{8}\right)-\left(4 \times 10^{8}\right) \), both have the same base and exponent, so subtract the coefficients: \(7 - 4 = 3\).
06

Write Result in Scientific Notation for Problem b

Express the subtracted coefficients in scientific notation: \(3 \times 10^{8}\).
07

Align Terms with Same Exponent for Problem d

Given \( \left(4 \times 10^{-12}\right)+\left(1 \times 10^{-12}\right) \), align the terms by simply adding the coefficients: \(4 + 1 = 5\).
08

Write Result in Scientific Notation for Problem d

Rewrite the result in scientific notation: \(5 \times 10^{-12}\).

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Key Concepts

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

Addition and Subtraction
When dealing with scientific notation, addition and subtraction problems become quite straightforward once you understand the basics. In scientific notation, numbers are expressed as the product of a coefficient and a power of 10. For example, \(5 \times 10^{-5}\).

To add or subtract these numbers:
  • First, ensure that the exponents (the powers of 10) are the same.
  • If they are not, you'll need to adjust the numbers so that they match, but in our original exercise, they already do.
  • Once the exponents are aligned, simply add or subtract the coefficients. The power of 10 remains unchanged.
This step may sound simple, but it鈥檚 crucial for solving these math problems accurately. Once you have your answer, you should express it in scientific notation, which helps in keeping the number concise and readable.
Coefficients
In scientific notation, a coefficient is the number that is multiplied by a power of 10. For instance, in \(7 \times 10^{8}\), the coefficient is 7.

It's important to remember when adding or subtracting in scientific notation that:
  • You only perform the operation on the coefficients.
  • The exponent, or the power of 10, remains consistent throughout the process as long as it was initially matching.
  • By focusing on changing just the coefficient, you streamline the process and reduce potential errors.
This is part of why scientific notation can be such a powerful tool in mathematics, especially for very large or very small numbers that would otherwise be cumbersome to write out or calculate.
Exponents
Exponents in scientific notation indicate how many times the base (which is usually 10) is to be multiplied by itself. For instance, in \(7 \times 10^{8}\), the exponent is 8.

When working with scientific notation, especially in addition or subtraction:
  • Ensure exponents are the same before performing any operations.
  • This consistency lets you focus solely on the coefficients, knowing the exponent won't alter the operation.
  • If exponent alignment is necessary, consider converting one number so that both have identical exponents before proceeding with addition or subtraction.
Understanding exponents is key to multiplying or dividing in scientific notation accurately, but with addition and subtraction, it simplifies to an alignment check.
Educational Math Problems
Educational math problems involving scientific notation are a fantastic way to sharpen skills in handling large and tiny numbers. They teach students to manage numbers with precision and simplify complex calculations. Let's break down why these problems are beneficial:
  • They enforce discipline in aligning exponents before performing operations on the coefficients.
  • These problems also help students form connections between various mathematical concepts and real-world applications, such as in physics or engineering.
  • By repeatedly practicing with scientific notation, students learn to translate complex, oversized numbers into a more manageable form.
Such exercises strengthen foundational math skills, which serve well into advanced studies and practical problem-solving scenarios.

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Most popular questions from this chapter

Write two conversion factors for each of the following. a. a 16\(\%\) (by mass) salt solution b. a density of 1.25 \(\mathrm{g} / \mathrm{mL}\) c. a speed of 25 \(\mathrm{m} / \mathrm{s}\)

Gold A troy ounce is equal to 480 grains, and 1 grain is equal to 64.8 milligrams. If the price of gold is $560 per troy ounce, what is the cost of 1 g of gold?

An object with a mass of 7.5 \(\mathrm{g}\) raises the level of water in a graduated cylinder from 25.1 \(\mathrm{mL}\) to 30.1 \(\mathrm{mL}\) . What is the density of the object?

Challenge Express each quantity in regular notation along with its appropriate unit. a. \(3.60 \times 10^{5} \mathrm{s} \quad\) b. \(5.4 \times 10^{-5} \mathrm{g} / \mathrm{cm}^{3}\) \(\quad\) c. \(5.060 \times 10^{3} \mathrm{km} \quad\) d. \(8.9 \times 10^{10} \mathrm{Hz}\)

Which number will produce the same number when rounded to three significant figures: 3.456,3.450. or 3.448?
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