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Chapter 7: Problem 7

Convert each number into scientific notation. $$7,000,000$$

Short Answer

Expert verified
7,000,000 is written as \( 7.0 \times 10^6 \) in scientific notation.

Step by step solution

01

Understand Scientific Notation

Scientific notation expresses a number as a product of a number between 1 and 10 and a power of 10. It is written in the form: \[ a \times 10^n \]where 1 鈮� a < 10 and n is an integer.
02

Identify the Coefficient

Move the decimal point in the number 7,000,000 to the left until only one non-zero digit remains to the left. For 7,000,000, the decimal point is initially at the end (7,000,000.0), moving it 6 places to the left gives us 7.0.
03

Determine the Exponent

Count the number of places the decimal point was moved. For 7,000,000, the decimal point was moved 6 places to the left. Therefore, the exponent is 6.
04

Combine the Coefficient and Exponent

Write the number in scientific notation by combining the coefficient (7.0) and the exponent (6) as \[ 7.0 \times 10^6 \].

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Key Concepts

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

Coefficients
A coefficient is a key part of scientific notation. It is the number between 1 and 10, used to represent the significant figures of a given number. For example, in the number 7,000,000, the coefficient is found by moving the decimal point to the left until there's only one non-zero digit to the left. We start with 7,000,000.0, and by moving the decimal six places to the left, we get 7.0. This 7.0 is our coefficient. Remember, the coefficient must always be between 1 and 10.

Understanding the significance of the coefficient helps in representing large or small numbers compactly. Whenever you convert a number into scientific notation, identifying the coefficient is your first task.
Exponents
Exponents are what make scientific notation powerful. The exponent in scientific notation indicates how many places the decimal point has been moved. If the decimal point is moved to the left, the exponent is positive, indicating a large number. If it鈥檚 moved to the right, the exponent is negative, indicating a small number.

For 7,000,000, the decimal point is at the end (7,000,000.0). Moving it six places to the left places it after 7.0. Therefore, the exponent is 6. This step ensures that you properly scale your number using powers of ten. The final scientific notation for 7,000,000 becomes \[ 7.0 \times 10^6 \]. This exponents part of scientific notation helps in expressing the magnitude of the number clearly and concisely.
Decimal Points
The position of the decimal point is crucial in converting a number to scientific notation. In the context of scientific notation, the decimal point helps establish the coefficient. You move the decimal point to create a coefficient between 1 and 10.

For instance, in converting 7,000,000 to scientific notation, start with the decimal at the end of the number (7,000,000.0). Moving it six places to the left, we place it just after the 7, yielding 7.0. This movement of the decimal point not only determines the coefficient but is also counted to define the exponent.

Understanding the role of the decimal point is essential for accurately converting and interpreting scientific notation.

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Most popular questions from this chapter

Simplify each of the following expressions as completely as possible. Final answers should be expressed with positive exponents only. (Assume that all variables represent positive quantities.) $$7 \cdot 7^{0}$$

Multiply out each of the following. As you work out the problems, identify those exercises that are either a perfect square or the difference of two squares. $$(x+6)(x-2)$$

Convert each number into scientific notation. $$0.00000003$$

Do your computations using scientific notation. Atomic measurements or other very small distances are often measured in units called angstroms. One angstrom, which is written \(1 \AA\), is equal to \(0.00000001 \mathrm{cm} .\) Write this number in scientific notation.

Convert each number into scientific notation. $$0.07$$
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