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Chapter 7: Problem 44

Convert each number into standard notation. $$6.81 \times 10^{-2}$$

Short Answer

Expert verified
0.0681

Step by step solution

01

Identify Scientific Notation Components

The given number is in scientific notation: \(6.81 \times 10^{-2}\). It consists of a coefficient (6.81) and a power of ten (-2).
02

Understand the Power of Ten

The exponent on the 10 indicates how many places to move the decimal point. A negative exponent tells us to move the decimal to the left.
03

Move the Decimal Point

Move the decimal point 2 places to the left (the number given by the exponent). Starting with 6.81, moving two places to the left gives us 0.0681.

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Key Concepts

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

Standard Notation
Standard notation is the regular way of writing numbers without using exponents. This format relies on place value to express the number. For example, instead of writing a number as a product of a coefficient and power of ten, you write out the entire number.

In scientific notation, a number is written as a product of a decimal (called the coefficient, which must be between 1 and 10) and a power of ten. For instance, the number 6.81 in scientific notation is written as $$6.81 \times 10^{-2}$$. To convert this to standard notation, you'll need to manipulate the position of the decimal point based on the exponent.
Negative Exponent
Understanding the exponent is crucial in converting from scientific to standard notation. When you see a negative exponent (like \text{-2} in the example), it indicates how many places to move the decimal point to the left.

In \text{scientific notation}, the negative exponent tells us to take the coefficient and make it smaller by shifting the decimal point. Think of it like moving backward from the decimal point. If the exponent was positive, we would move the decimal to the right, making the number larger.

For example, $$6.81 \times 10^{-2}$$ means we need to move the decimal two places to the left to convert it into standard notation.
Decimal Point Movement
To convert scientific notation to standard notation, moving the decimal point correctly is key. Let's break it down with an example.

Given the number $$6.81 \times 10^{-2}$$, you need to move the decimal point two places to the left due to the \text{-2} exponent. Here鈥檚 a step-by-step way to do it:
  • Start with the number 6.81.
  • Since the exponent is \text{-2}, you move the decimal point two positions to the left.
  • The first move places the decimal point in front of 6, leading to 0.681.
  • Move one more position to the left to place the decimal point in front of 681, resulting in 0.0681.

So, the standard notation of $$6.81 \times 10^{-2}$$ is 0.0681. Always remember that the exponent tells you the direction and number of places to move the decimal point.

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