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Chapter 10: Problem 51

For Exercises \(49-64\), convert to standard form. $$8 \times 10^{-2}$$

Short Answer

Expert verified
The standard form is 0.08.

Step by step solution

01

Identify the Given Number

The given number is in scientific notation: $$8 \times 10^{-2}$$
02

Understand Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small in a compact form. The given number has a coefficient (8) and an exponent (-2).
03

Convert Negative Exponent

A negative exponent means dividing by that power of 10. So, $$8 \times 10^{-2}$$ is equivalent to $$8 / (10^2)$$
04

Perform the Division

Calculate the division: $$8 / 100 = 0.08$$

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Key Concepts

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

Standard Form
When we talk about standard form, we are referring to the more common way of writing numbers, without exponents. This format makes it easy to read and understand numbers at a glance. For instance, instead of writing the number as a product of a coefficient and a base of 10, we write it out fully. In the case of the exercise, the number given was in scientific notation: \(8 \times 10^{-2}\). Let's break it down: in standard form, this becomes 0.08. By converting it, we can easily see that 0.08 is simply eight hundredths.
Negative Exponent
One crucial aspect of scientific notation is understanding negative exponents. A negative exponent indicates that the base (in this case, 10) is divided, rather than multiplied. Here's how to interpret it: \(10^{-2}\) translates to dividing by 100, because moving the decimal point two places to the left changes the number into a smaller value. Think of it this way: instead of multiplying 8 by a large number, we're making it smaller by converting \(8 \times 10^{-2}\) to \(8 / (10^2)\), which results in 0.08.
Division
Division is a key operation in converting scientific notation with negative exponents. In our example, the exponent \(-2\) becomes \(10^2\) and instead of multiplying, we divide. To break this down: \(8 \div 100 = 0.08\).

Here鈥檚 a simple way to think about it:
  • Identify the base and the exponent.
  • If the exponent is negative, rewrite it as a division by that power of 10.
  • Perform the actual division operation.
When you divide 8 by 100, you move the decimal two places to the left, giving you 0.08. This is how negative exponents result in smaller numbers.
Scientific Notation
Scientific notation is a handy way of expressing very large or very small numbers succinctly. The format involves a coefficient and a power of 10. This notation greatly simplifies calculations and readability, especially in scientific and engineering contexts. The exercise gave us the number in scientific notation: \(8 \times 10^{-2}\), which, when converted, helps in easily understanding that it's equal to 0.08. Remember:
  • A positive exponent moves the decimal point to the right.
  • A negative exponent moves it to the left.
This compact way of writing numbers saves space and makes it quicker to perform mathematical operations.

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Most popular questions from this chapter

Multiply the polynomials. $$\begin{array}{r}y^{2}-7 y+3 \\\\\times \quad 3 y-5 \\\\\hline\end{array}$$

The cost (in dollars) to rent a car is given by the polynomial \(29.99 x+0.40 y .\) In this context, \(x\) is the number of days that the car is rented and \(y\) is the number of miles driven. a. Evaluate the polynomial for \(x=3\) and \(y=350 .\) Interpret the answer in the context of this problem. b. Evaluate the polynomial for \(x=7\) and \(y=720 .\) Interpret the answer in the context of this problem.

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