Chapter 2: Problem 49
Convert the following exponential numbers to scientific notation. (a) \(352 \times 10^{4}\) (b) \(0.191 \times 10^{-5}\)
Short Answer
Expert verified
(a) \(3.52 \times 10^6\); (b) \(1.91 \times 10^{-6}\).
Step by step solution
01
Understanding the Concept
Scientific notation is a way of expressing numbers as a product of a number between 1 and 10 and a power of 10. This format is useful for simplifying numbers that are either very large or very small.
02
Convert (a) to Scientific Notation
For the number 352, move the decimal point to the left until you have a number between 1 and 10. This means moving the decimal point 2 places to the left, transforming 352 into 3.52. The exponent on 10 will increase by the same number of places that the decimal was moved, contributing to the existing exponent: 352 can be written as 3.52 with the decimal point moved 2 places.Hence, the expression becomes:\(352 \times 10^4 = 3.52 \times 10^{4+2} = 3.52 \times 10^6\).
03
Convert (b) to Scientific Notation
For the number 0.191, you need to move the decimal point to the right to create a number between 1 and 10. This involves moving the decimal point 1 place to the right, changing it to 1.91. The power of 10 will decrease by 1 for each place moved.0.191 can be written as 1.91 with the decimal point moved 1 place.Therefore, the expression becomes:\(0.191 \times 10^{-5} = 1.91 \times 10^{-5-1} = 1.91 \times 10^{-6}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Exponents
Exponents are a fundamental concept in mathematics, especially when dealing with large or small numbers. An exponent is a small number written above and to the right of a base number. It tells you how many times to multiply the base number by itself. For example, in the expression \(10^4\), the number 10 is the base and 4 is the exponent. This means you multiply 10 by itself 4 times, resulting in 10,000.
Exponents are often used in scientific notation to condense very large or very small numbers. This simplifies calculations and makes numbers easier to read and understand. When you see a number with an exponent, it's essentially showing repeated multiplication, which is extremely helpful when expressing quantities in scientific fields.
In our exercise, we convert numbers like 352 or 0.191 into a product of a number between 1 and 10 and a power of 10 (i.e., a base of 10 raised to an exponent). This transformation is the essence of scientific notation.
Exponents are often used in scientific notation to condense very large or very small numbers. This simplifies calculations and makes numbers easier to read and understand. When you see a number with an exponent, it's essentially showing repeated multiplication, which is extremely helpful when expressing quantities in scientific fields.
In our exercise, we convert numbers like 352 or 0.191 into a product of a number between 1 and 10 and a power of 10 (i.e., a base of 10 raised to an exponent). This transformation is the essence of scientific notation.
The Role of the Decimal Point
The decimal point plays a crucial role in converting numbers into scientific notation. It's essentially a dot that separates the whole part of a number from its fractional part. Understanding how to move the decimal point correctly is key to mastering scientific notation.
When converting numbers like 352 into scientific notation, we move the decimal two places to the left to get 3.52. This moves our number into a form where the decimal is between 1 and 10, which is essential for proper scientific notation. The movement of the decimal point dictates how much we adjust the exponent of the power of 10. If moving the decimal left, we increase the exponent; if right, we decrease it.
On the other hand, for small numbers like 0.191, we move the decimal point one place to the right to form 1.91. This reversal is required to make sure that our number is again between 1 and 10, maintaining the standard format of scientific notation and adjusting the corresponding exponent as needed. By mastering the control of the decimal point, you can easily convert any number into this clearly defined format.
When converting numbers like 352 into scientific notation, we move the decimal two places to the left to get 3.52. This moves our number into a form where the decimal is between 1 and 10, which is essential for proper scientific notation. The movement of the decimal point dictates how much we adjust the exponent of the power of 10. If moving the decimal left, we increase the exponent; if right, we decrease it.
On the other hand, for small numbers like 0.191, we move the decimal point one place to the right to form 1.91. This reversal is required to make sure that our number is again between 1 and 10, maintaining the standard format of scientific notation and adjusting the corresponding exponent as needed. By mastering the control of the decimal point, you can easily convert any number into this clearly defined format.
Power of 10 in Scientific Notation
The power of 10 is what enables scientific notation to express extremely large or small numbers concisely. The power of 10 is denoted by an exponent, which represents how many times to multiply the number 10 by itself. This is especially useful in fields like physics, chemistry, and engineering.
In scientific notation, a number is written as the product of two components: a decimal (between 1 and 10) and a power of 10. In the example \(352 \times 10^4\), converting it to scientific notation involves moving the decimal two places to form 3.52, which alters the exponent from 4 to 6, resulting in \(3.52 \times 10^6\). This increase reflects that we moved the decimal towards the left for larger numbers.
Similarly, when dealing with small numbers like 0.191, moving the decimal helps adjust the power of 10 downward. This conversion, \(0.191 \times 10^{-5}\) becomes \(1.91 \times 10^{-6}\) as the fraction increases and the exponent adjusts accordingly. Thus, the power of 10 serves as the backbone for effectively scaling numbers up or down, making calculations both manageable and understandable.
In scientific notation, a number is written as the product of two components: a decimal (between 1 and 10) and a power of 10. In the example \(352 \times 10^4\), converting it to scientific notation involves moving the decimal two places to form 3.52, which alters the exponent from 4 to 6, resulting in \(3.52 \times 10^6\). This increase reflects that we moved the decimal towards the left for larger numbers.
Similarly, when dealing with small numbers like 0.191, moving the decimal helps adjust the power of 10 downward. This conversion, \(0.191 \times 10^{-5}\) becomes \(1.91 \times 10^{-6}\) as the fraction increases and the exponent adjusts accordingly. Thus, the power of 10 serves as the backbone for effectively scaling numbers up or down, making calculations both manageable and understandable.
