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Chapter 10: Problem 17

For Exercises \(13-20,\) write each power of 10 in exponential notation. $$0.001$$

Short Answer

Expert verified
0.001 in exponential notation is \(10^{-3}\).

Step by step solution

01

Identify the number as a power of 10

Given the number is 0.001. Recognize that this number is a fraction and can be expressed as a power of 10.
02

Convert the number into fraction form

0.001 can be written as the fraction \(\frac{1}{1000}\).
03

Express the denominator as a power of 10

Notice that 1000 is equivalent to \(10^3\). Therefore, \(\frac{1}{1000}\) can be rewritten as \(\frac{1}{10^3}\).
04

Simplify the expression into exponential notation

Using the property of exponents, \(\frac{1}{10^3}\) can be rewritten as \(10^{-3}\).

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Key Concepts

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

Powers of Ten
Understanding powers of ten is crucial when working with exponential notation. Powers of ten are a way to express large or small numbers succinctly. Each power of ten represents how many times 10 is multiplied by itself. For example:
  • 10^1 = 10
  • 10^2 = 100
  • 10^3 = 1,000
Conversely, we can also express very small numbers with negative powers of ten. The more negative the exponent, the smaller the number. For example:
  • 10^{-1} = 0.1
  • 10^{-2} = 0.01
  • 10^{-3} = 0.001
Powers of ten make reading, writing, and calculating with large and small numbers easier.
Fraction to Exponent
When converting a fraction to an exponent involving base 10, it鈥檚 helpful to understand that a fraction like \(\frac{1}{10^n}\) can be written as 10 to a negative power. For example, let's take the fraction \(\frac{1}{1000}\):
  • First, recognize that the denominator (1000) is a power of 10.
  • In this case, 1000 is equivalent to 10 raised to the power of 3, so we have \(\frac{1}{10^3}\).
  • Applying the property of exponents, \(\frac{1}{10^3}\) is the same as 10 raised to the power of -3.
By converting the denominator to a power of ten and using the rule for negative exponents, we simplify \(\frac{1}{10^n}\) to 10^{-n}. This method allows us to change fractions into a format that's easier to work with in equations.
Negative Exponents
Negative exponents can be confusing at first, but they are very useful for representing small numbers. Here's how they work:
  • When an exponent is negative, it indicates a reciprocal. For example, 10^{-n} means the reciprocal of 10^n.
  • This means \(10^{-3}\) is equivalent to \(\frac{1}{10^3} = \frac{1}{1000}\).
  • Negative exponents effectively shift the position of the decimal point to the left. For instance: \(10^{-1} = 0.1 \), \(10^{-2} = 0.01\), and \(10^{-3} = 0.001\).
Understanding negative exponents is crucial for grasping the concept of very small numbers. This knowledge is particularly useful in scientific notation, where large and small values are often represented concisely using powers of ten.

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Most popular questions from this chapter

For Exercises \(33-48,\) convert to scientific notation. $$0.58$$

Add the polynomials. $$\left(-2 a^{2}+5 a-3\right)+\left(a^{2}-11 a-7\right)$$

Factor out the greatest common factor. $$24 m^{4} n^{2}+36 m^{3} n-48 m^{2} n^{3}$$

Perform the indicated operations. $$\left(-t+8 t^{3}-3 t^{4}\right)-\left(5 t+4 t^{4}+6 t^{5}\right)+\left(15 t^{3}+2 t^{4}+6 t^{5}\right)$$

For Exercises \(49-64\), convert to standard form. $$9.001 \times 10^{8}$$
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