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Chapter 5: Problem 83

Write each number in decimal notation without the use of exponents. $$3.4 \times 10^{9}$$

Short Answer

Expert verified
The number \(3.4 \times 10^{9}\) in decimal notation is 3400000000.

Step by step solution

01

Understand Scientific Notation

First, understand how scientific notation works. A number in scientific notation is written as the product of a coefficient and a power of 10. In the case of \(3.4 \times 10^{9}\), 3.4 is the coefficient and 9 is the exponent. This means that the decimal point in 3.4 needs to be moved 9 places to the right.
02

Move the Decimal Point

Since the exponent is 9, which is a positive number, move the decimal point in the coefficient (3.4) 9 places to the right. This means adding nine zeros after the number 3.4.
03

Write in Decimal Notation

After moving the decimal point 9 places to the right and filling the gaps with zeros, the number should be written as 3400000000.

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Key Concepts

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

Understanding Decimal Notation
Decimal notation is a straightforward way of writing numbers using the base ten system. Unlike in scientific notation, where numbers are expressed as the product of a coefficient and a power of ten, decimal notation represents numbers in their plain, expanded form.

For example, when we have a number like \(3.4 \times 10^{9}\), the task is to convert it into decimal notation. This means rewriting it without using the exponent, shifting the decimal point directly to reflect the actual value of the number.
  • Decimal notation makes it clear to understand and interpret large or small amounts by showing every number directly.
  • Increases ease of computation and clarity in daily use as seen with currency amounts or population figures.

Overall, understanding decimal notation helps you visualize numbers more intuitively, and simplifies mathematical operations involving them.
Role of Exponents in Numbers
Exponents play a crucial role in mathematics as they provide a compact and efficient way to represent very large or very small numbers. An exponent indicates how many times a number (the base) is multiplied by itself.

For example, in scientific notation, \(10^{9}\) means that 10 is multiplied by itself 9 times, which results in a billion (1,000,000,000).
  • Exponents save space by allowing large values to be written succinctly.
  • They simplify complex calculations by breaking down multiplication.

By understanding exponents, you can easily move between different forms of notation, like converting \(3.4 \times 10^{9}\) to a more accessible decimal form. This skill is valuable, not only in mathematics itself but also in fields like science and engineering, where large numbers frequently occur.
The Importance of Mathematics Education
Mathematics education is critical because it provides students with the tools they need to understand and manipulate numbers effectively. Learning concepts like scientific notation and exponents not only aids in solving practical mathematical problems but also enhances logical thinking and problem-solving skills.

Mathematical proficiency allows individuals to access higher learning and career opportunities that require analytical thinking.
  • It builds a foundation for understanding real-world phenomena such as space exploration and data analysis.
  • Encourages mental discipline and the development of structured thinking processes.

Through continuous learning and application, students can master mathematical concepts like converting scientific notation to decimal notation, ensuring they are well-prepared for advanced studies and diverse professional fields.

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Most popular questions from this chapter

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There are many exponential expressions that are equal to \(36 x^{12},\) such as \(\left(6 x^{6}\right)^{2},\left(6 x^{3}\right)\left(6 x^{9}\right), 36\left(x^{3}\right)^{9},\) and \(6^{2}\left(x^{2}\right)^{6}\)

Find the missing factor. $$(\quad ) \left(-\frac{1}{4} x y^{3}\right)=2 x^{5} y^{3}$$

In Exercises \(85-86,\) the variable \(n\) in each exponent represents a natural Number. Divide the polynomial by the monomial. Then use polynomial multiplication to check the quotient. $$\frac{12 x^{15 n}-24 x^{12 n}+8 x^{3 n}}{4 x^{3 n}}$$

Exercises \(110-112\) will help you prepare for the material covered in the next section. In each exercise, perform the long division without using a calculator, and then state the quotient and the remainder. $$. 9 8 \longdiv { 2 5 . 1 8 7 }$$

Perform the indicated operations. $$(x+4)(x-5)-(x+3)(x-6)$$
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