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Chapter 2: Problem 34

Write each number in scientific notation. -0.00000022

Short Answer

Expert verified
The scientific notation for -0.00000022 is \(-2.2 \times 10^{-7}\).

Step by step solution

01

Determine the exponent

To shift the decimal in -0.00000022 so that it becomes a number between 1 and 10, we need to count the number of times we need to move the decimal point. Starting from its current position, we move it seven times to the right until we have -2.2.
02

Write the number in scientific notation

Now that we've moved the decimal place 7 times to the right, we can express -0.00000022 in scientific notation. Since we moved the decimal 7 places to the right, our exponent for 10 will be -7. So, the scientific notation for -0.00000022 is \(-2.2 \times 10^{-7}\).

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Key Concepts

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

Negative Exponents
Negative exponents are a crucial part of understanding scientific notation, particularly when dealing with small numbers. When the exponent is negative, it indicates that the decimal point has been moved to the right when writing a number in its standard form.
For example, if the exponent is -7 in a scientific notation, as in \(-2.2 \times 10^{-7}\), it means that the decimal point has been shifted 7 places from left to right to reach the proper format.
  • A negative exponent signifies a division by 10.
  • Every negative move corresponds to moving the decimal left in the original number.
Understanding negative exponents allows you to interpret very small numbers compactly.
Decimal Placement
Decimal placement is key in converting numbers to scientific notation. This process determines how we shift the position of the decimal point to express the number in a more manageable form.
In the example -0.00000022, the decimal needed to be moved to create a number between 1 and 10, resulting in -2.2.
  • Counting the necessary places moved gives the exponent size, here it was 7 places.
  • This shift displays how tiny the original number is when expressed more simply.
Always count the steps taken to get the decimal within the appropriate boundary, ensuring correct scientific notation.
Standard Form
Standard form, often used interchangeably with scientific notation, is a method of writing numbers to make them easier to handle.
Specifically, it expresses numbers as a product of a number between 1 and 10 and a power of ten.
In the provided example, -0.00000022 becomes -2.2 x 10^-7 when written in standard or scientific form.
  • The initial number becomes manageable by reducing zeros.
  • Using powers of ten to reflect the decimal's original position simplifies representation.
This form is particularly beneficial in scientific calculations for easier computation and clarity.
Algebra Basics
Algebra basics underpin the conversion process to scientific notation, including manipulations involving exponents and decimals.
Understanding these basics expands one's ability to solve problems efficiently by recognizing patterns and applying rules like exponentiation.
  • Scientific notation itself is an application of algebra, using multiplication and exponent rules.
  • An alaphemerization of numbers makes complex equations more approachable.
Being fluent in these foundational algebra concepts ensures ease and accuracy when dealing with such transformations in numbers.

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