Problem 19 For Exercises $13-20,$ write e... [FREE SOLUTION]

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Prealgebra

Julie Miller, Molly O鈥橬eill, Nancy Hyde

$Math Studyset 91影视 Explanations$ Math

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Chapter 10: Problem 19

For Exercises $13-20,$ write each power of 10 in exponential notation. $$0.0001$$

Short Answer

Expert verified

The number 0.0001 in exponential notation is $10^{-4}$.

Step by step solution

- Identify the Place Value

First, identify the place value of the digit 1 in the decimal number 0.0001. Here, 1 is in the fourth position to the right of the decimal point.

- Determine the Exponent

Since 1 is four positions to the right of the decimal point, the exponent in the power of 10 will be negative. Count the number of decimal places which is 4, thus the exponent should be -4.

- Write in Exponential Notation

Write the number in exponential notation using the determined exponent. Therefore, 0.0001 is written as $10^{-4}$.

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Key Concepts

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

powers of 10

Understanding powers of 10 is crucial when working with exponential notation. The powers of 10 are the basis for expressing large or small numbers compactly.
When you see a number like 10 raised to the power of 2, it means multiplying 10 by itself twice:
\[ 10^2 = 10 \times 10 = 100 \]
This can work in the opposite direction too. If you see 10 raised to the power of -2, it means you take the reciprocal of 10 squared:
\[ 10^{-2} = \frac {1}{10^2} = \frac {1}{100} = 0.01 \]
Powers of 10 simplify arithmetic operations, especially multiplication and division, through shifting the decimal point.

place value

Place value denotes the numerical value that a digit has by its position in a number. For the decimal number 0.0001, the digit 1 is in the ten-thousandth place.
Every position to the right of the decimal point represents a fraction, decreasing by a power of ten. For example:
鈥� First place: tenths, \[10^{-1} = 0.1\]
鈥� Second place: hundredths, \[10^{-2} = 0.01\]
鈥� Third place: thousandths, \[10^{-3} = 0.001\]
鈥� Fourth place: ten-thousandths, \[10^{-4} = 0.0001\]
Understanding place value helps determine the exponent for powers of 10 in exponential notation.

negative exponents

Negative exponents represent the reciprocal of the number with a positive exponent. For instance,
鈥� \[10^{-1} = \frac{1}{10^1} = 0.1\]
鈥� \[10^{-2} = \frac{1}{10^2} = 0.01\]
鈥� \[10^{-3} = \frac{1}{10^3} = 0.001\]
鈥� And similarly, \[10^{-4} = \frac{1}{10^4} = 0.0001\]
Negative exponents simplify the representation of very small numbers. Instead of writing many zeros, you convey the same value with a compact notation, like \[10^{-4}\] instead of 0.0001.

decimal numbers

Decimal numbers are numbers that have a whole number part and a fractional part, separated by a decimal point.
Take the number 0.0001 as an example. It has:
鈥� A whole number part: 0
鈥� A fractional part: 0001
Each digit in the fractional part has a value based on its place. In this case, the 1 is in the ten-thousandth place. Decimal numbers can be efficiently converted to exponential notation by using powers of 10, making complex calculations easier to handle.

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91影视

For Exercises \(13-20,\) write each power of 10 in exponential notation. $$0.0001$$

Short Answer

Step by step solution

- Identify the Place Value

- Determine the Exponent

- Write in Exponential Notation

Key Concepts

powers of 10

place value

negative exponents

decimal numbers

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