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Chapter 0: Problem 67

Write each number in decimal notation without the use of exponents. $$6 \times 10^{-4}$$

Short Answer

Expert verified
The number \(6 \times 10^{-4}\) can be written in decimal notation as 0.0006

Step by step solution

01

Understand the number

The original number is \(6 \times 10^{-4}\). This number is in scientific notation, and the task is to write it in decimal notation, moving the decimal point to the left by four places, as indicated by the power of -4.
02

Change the position of the decimal

Begin at the location of the decimal in the number 6. Since there is no decimal shown, it is understood to be at the end of the number. Moving the decimal point four places to the left results in 0.0006.
03

Write The Result

After moving the decimal four places to the left, the number in decimal notation is 0.0006. Hence, \(6 \times 10^{-4}\) written in decimal notation without using exponents is 0.0006.

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Key Concepts

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

Decimal Notation
Decimal notation is a system of writing numbers that uses a decimal point to separate the whole number part from the fractional part. It's the most commonly used method to represent numbers in daily life, and in mathematics, it allows us to comprehend the value of numbers at a glance.

For example, when you see the number 0.0006 in decimal notation, it tells you immediately that the value is six ten-thousandths. No exponents are needed here; the placement of the decimal point gives you all the information about the magnitude of the number.
Negative Exponent
A negative exponent indicates that the number is a fraction and represents division by the base raised to the corresponding positive exponent. In simple terms, when you see a negative exponent, you need to 'flip' the base to the denominator and change the negative exponent to a positive one.

For instance, imagine you have the expression \(10^{-4}\). A negative exponent means the number is essentially \(1/10^4\), which can be written as \(1/10000\) or 0.0001. This 'flipping' and changing the sign of the exponent from negative to positive is crucial when converting scientific notation to decimal notation.
Scientific Notation
Scientific notation is a way to express very large or very small numbers succinctly. It鈥檚 especially useful in sciences for handling the vast range of measurements. In scientific notation, a number is written as the product of a number between 1 and 10 and a power of 10.

Take for example \(6 \times 10^{-4}\). This notation tells you that you're dealing with 6 times a power of ten, with the exponent indicating how many places to move the decimal point. If the exponent is negative, move the decimal to the left. If positive, move it to the right. Understanding scientific notation is crucial because it is widely used in calculating and reporting scientific data.
Place Value
Place value is about the position of a digit in a number and how that position dictates the value of that digit. In the decimal system, every place to the left of the decimal increases by a power of 10, and every place to the right of the decimal decreases by a power of 10.

Let's explore 0.0006 once more. The 6 is in the ten-thousandths place, which is four places to the right of the decimal point. In other words, its place value is \(6 \times 1/10000\) or \(6 \times 10^{-4}\). Every time you move one place to the left, the value of the digit becomes ten times greater.

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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. Although \(20 x^{3}\) appears in both \(20 x^{3}+8 x^{2}\) and \(20 x^{3}+10 x\) I'll need to factor \(20 x^{3}\) in different ways to obtain each polynomial's factorization.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. The model \(P=-0.18 n+2.1\) describes the number of pay phones, \(P,\) in millions, \(n\) years after \(2000,\) so I have to solve a linear equation to determine the number of pay phones in 2010

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells blank audiocassette tapes. The weekly fixed cost is \(\$ 10,000\) and it costs \(\$ 0.40\) to produce each tape. The selling price is \(\$ 2.00\) per tape. How many tapes must be produced and sold each week for the company to generate a profit?

Describe the solution set of \(|x|>-4\)

Your local electronics store is having an end-of-the-year sale. The price on a plasma television had been reduced by \(30 \%\) Now the sale price is reduced by another \(30 \% .\) If \(x\) is the television's original price, the sale price can be modeled by $$(x-0.3 x)-0.3(x-0.3 x)$$ a. Factor out \((x-0.3 x)\) from each term. Then simplify the resulting expression. b. Use the simplified expression from part (a) to answer these questions. With a \(30 \%\) reduction followed by a \(30 \%\) reduction, is the television selling at \(40 \%\) of its original price? If not, at what percentage of the original price is it selling?
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