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Chapter 4: Problem 38

Write each mumber in standard notation. See Example \(2 .\) \(38 .-6 \times 10^{-4}\)

Short Answer

Expert verified
-0.0006

Step by step solution

01

Understand the Scientific Notation

The number given is in scientific notation: \( -6 \times 10^{-4} \). Scientific notation expresses numbers as a product of a coefficient and a power of 10.
02

Interpret the Power of 10

The exponent in the power of 10 tells us how many places to move the decimal point. A negative exponent indicates that the decimal should be moved to the left. Here, \( 10^{-4} \) indicates 4 places to the left.
03

Move the Decimal Point

Start with the coefficient \( -6 \). Move the decimal 4 places to the left to convert it to standard notation: \[ -6 \rightarrow -0.0006 \]

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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 a way of writing numbers without using exponents. This method makes it simpler to read and understand values, especially when dealing with very large or very small numbers.
In standard notation, you write out the entire number as it is, without indicating powers of 10. This contrasts with scientific notation, which uses a coefficient and an exponent to represent large or small values succinctly.
For instance, in the example, converting \( -6 \times 10^{-4} \) to standard notation involves moving the decimal point to give you \[ -0.0006 \]. This direct representation can be easier to grasp.
Negative Exponent
In mathematics, an exponent indicates how many times to multiply a base number by itself. A negative exponent, however, indicates how many times to divide by the base number, rather than multiply.
When you see \( 10^{-4} \), it means you need to divide 10 four times, or move the decimal point in the number 4 places to the left. This is crucial for converting numbers from scientific notation to standard notation.
So, for \( -6 \times 10^{-4} \):
  • Start with the number -6
  • Move the decimal 4 places to the left.
This shifts the number to \(-0.0006\), effectively making a much smaller value.
Decimal Point
The decimal point is a dot used to separate the integer part of a number from its fractional part. Understanding its position is essential for performing arithmetic operations and converting scientific notation to standard notation.
For instance, in the problem \( -6 \times 10^{-4} \), you start with the integer -6, which can be seen as \(-6.0\).
To convert this to standard notation, you move the decimal point. Moving it four places to the left changes -6.0 into \(-0.0006\). This step is critical for accurately understanding and representing numbers, especially those conveyed in scientific notation.

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Most popular questions from this chapter

Find each product. $$ (7-2 x)(7+2 x) $$

Simplify by combining like terms whenever possible. Write results that have more than one term in descending powers of the variable. \(-3 a^{8}+4 a^{8}-3 a^{2}\)

If a negative number is raised to an even power, the result is positive. For example, $$(-3)^{4}=(-3)(-3)(-3)(-3)=+81-\text { that is, } 81$$ If a negative number is raised to an odd power, the result is negative. For example, $$(-3)^{5}=(-3)(-3)(-3)(-3)(-3)=-243$$ Without actually performing the computation, state whether the power is positive or negative. (a) \((-4)^{4}(-4)^{2}\) (b) \((-4)^{3}(-4)^{8}\) (c) \((-4)^{12}(-4)^{16}\) (d) (-4)\((-4)^{96}\)

Find each product. $$ (2 m+5)(2 m-5) $$

Find each product. $$ 2 m(3 m+2) $$
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