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Chapter 0: Problem 74

Write each number in decimal notation without the use of exponents. $$-3.14 \times 10^{-3}$$

Short Answer

Expert verified
-0.00314

Step by step solution

01

Identify the parts of the number in scientific notation

The given number is \(-3.14 \times 10^{-3}\). Here, -3.14 is the coefficient, and -3 is the exponent.
02

Shift the decimal in the coefficient

The exponent of -3 means to move the decimal point three places to the left in the coefficient.
03

Write the number in decimal notation

After shifting the decimal point three places to the left, the number in decimal notation is -0.00314.

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Key Concepts

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

Scientific Notation
Scientific notation is a way to express very large or very small numbers in a compact form. It represents numbers as a product of a coefficient and a power of ten.
The coefficient is typically a number between 1 and 10, whereas the exponent on the ten signifies how many times to multiply or divide the coefficient by ten. For example, in the expression \(-3.14 \times 10^{-3}\), \(-3.14\) is the coefficient, and \(10^{-3}\) shows that the decimal is moved three places to the left.
  • Scientific notation is useful for simplifying calculations with large or small numbers.
  • It's commonly used in scientific and engineering contexts where precision with large numbers is essential.
In summary, scientific notation helps manage numbers efficiently without dealing directly with cumbersome zeros. This method is very handy for both computation and communication.
Exponents
Exponents are a shorthand notation used to express repeated multiplication of a number by itself. In scientific notation, they are the power of ten that explains how many times to multiply or divide the coefficient by ten.
For example, in \(10^{-3}\), the exponent is \(-3\). This means that you need to divide the coefficient by 10 three times, or equivalently, move the decimal three places to the left.
  • Positive exponents move the decimal point to the right, increasing the number's size.
  • Negative exponents move the decimal point to the left, decreasing the number's size.
Understanding exponents is key as it directly relates to how we manipulate the decimal in scientific notation calculations. Mastering exponents allows for smoother transitions between different forms of number expressions.
Decimal Point Shifting
Decimal point shifting is a method used to convert numbers from scientific notation to decimal notation. It involves moving the decimal point in the coefficient based on the exponent's value.
In the expression \(-3.14 \times 10^{-3}\), the exponent \(-3\) indicates that the decimal should be moved three places to the left. This adjustment changes the original coefficient \(-3.14\) to \(-0.00314\).
  • For a negative exponent, the decimal point shifts to the left. This makes the number smaller.
  • For a positive exponent, the decimal point shifts to the right. This enlarges the number.
Decimal point shifting is a practical skill. It bridges the gap between scientific notation and everyday decimal notation. Once understood, transforming numbers between forms becomes straightforward and intuitive.

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Most popular questions from this chapter

Explain how to solve \(x^{2}+6 x+8=0\) using factoring and the zero-product principle.

Solve each equation. $$\sqrt{x+8}-\sqrt{x-4}=2$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. An elevator at a construction site has a maximum capacity of 3000 pounds. If the elevator operator weighs 245 pounds and each cement bag weighs 95 pounds, how many bags of cement can be safely lifted on the elevator in one trip?

Find all integers \(b\) so that the trinomial can be factored. $$x^{2}+b x+15$$

This will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right)$$
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