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Chapter 2: Problem 3

Determine if each number is in scientific notation. \(0.16 \times 10^{-4}\)

Short Answer

Expert verified
The given number \(0.16 \times 10^{-4}\) is not in scientific notation because, when rewritten in standard scientific notation, it becomes \(1.6 \times 10^{-5}\) which has a different coefficient and exponent.

Step by step solution

01

Rewrite the number in decimal form

To check the format of the given number, let's first rewrite it in decimal form: \[ 0.16 \times 10^{-4} = 0.16 \div 10^4 = 0.16 \div 10000 = 0.000016 \]
02

Rewrite the decimal number in scientific notation

Now, let's rewrite the decimal number in standard scientific notation (a 脳 10^n, where 1 鈮� a < 10 and n is an integer): \[ 0.000016 = 1.6 \times 10^{-5} \]
03

Compare the original and rewritten numbers

Finally, let's compare the given number with the rewritten number in the standard scientific notation: \[ 0.16 \times 10^{-4} \ne 1.6 \times 10^{-5} \] Since the given number is not equal to the standard scientific notation (with 1 鈮� a < 10), we can conclude that the original number is not in scientific notation.

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Key Concepts

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

Decimal Form
Decimal form is a way of representing numbers that employs the base ten numeral system. It's a straightforward way to express numbers and is commonly used in everyday life. For instance, the number '117' is in decimal form. In this base ten system, each position represents a power of ten. To better understand decimal form, consider the number 0.000016:
  • The '1' is in the hundred-thousandths place, meaning it stands for \(1\times10^{-5}\).
  • The '6' is in the millionths place, making it \(6\times10^{-6}\).
Breaking down numbers into such parts helps in understanding place value, which is crucial for converting numbers into different forms like scientific notation.
Standard Form
In mathematics, the term 'standard form' can have multiple meanings depending on the context. But notably, in dealing with numbers, it often refers to the way numbers are expressed as plain numbers. When considering scientific notation, standard form refers to the expression of a number such that it consists of a coefficient \(a\) followed by 10 raised to an exponent \(n\):
  • The number should be expressed as \(a \times 10^n\).
  • The coefficient \(a\) should be greater than or equal to 1 and less than 10.
  • \(n\), the exponent, is an integer which denotes the number of times the base 10 is raised.
In the exercise above, the initial number given, 0.16 \( \times 10^{-4}\), is not in this "standard form" since 0.16 is less than 1.
Exponents
Exponents are an essential part of mathematics, representing repeated multiplication of a number by itself. In the expression \(10^4\), 10 is the base and 4 is the exponent. Exponents simplify expressions by reducing repeated multiplication into a more compact form. They also play a crucial role in scientific notation by moving the decimal point:
  • A positive exponent moves the decimal point to the right.
  • A negative exponent moves the decimal point to the left, which is the case with \(10^{-4}\) in our original problem.
Understanding exponents is crucial, as it aids in transitioning between decimal form and scientific notation effortlessly.
Mathematics Education
Education in mathematics focuses on strengthening both foundational and advanced concepts, including subjects like scientific notation, decimals, and exponents. Grasping these concepts is vital for students, as they lay the groundwork for more complex topics in various fields of science and engineering. Effective education strategies include:
  • Providing visuals and practical examples to clarify abstract concepts.
  • Encouraging problem-solving and critical thinking to apply what they learn.
  • Using real-life examples to make learning more relatable and engaging.
By focusing on these teaching methods, educators aim to make mathematical principles accessible and intuitive for students, facilitating a deeper understanding and the ability to apply these skills in real-world scenarios.

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Most popular questions from this chapter

Simplify using the quotient rule. $$\frac{200 x^{x} y^{3}}{20 x^{10} y^{11}}$$

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Write each number in scientific notation. 0.00418

Perform the operation as indicated. Write the final answer without an exponent. \(\frac{8.4 \times 10^{12}}{-7 \times 10^{9}}\)

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