Solve 0.0000003 × (9.1 × 10⁻¹⁵) In Standard Form A Step-by-Step Guide
In the realm of mathematics, particularly when dealing with extremely large or small numbers, scientific notation becomes an invaluable tool. This article delves into the process of multiplying numbers expressed in scientific notation, providing a comprehensive, step-by-step solution to the problem 0.0000003 × (9.1 × 10⁻¹⁵). We will explore the fundamental principles behind scientific notation, the rules of multiplication involving exponents, and how to express the final result in standard form. This exploration aims to equip readers with a solid understanding of these concepts, enabling them to tackle similar mathematical challenges with confidence.
Scientific Notation: A Primer
Scientific notation, also known as standard form, is a way of expressing numbers as a product of a number between 1 and 10 (inclusive of 1 but exclusive of 10) and a power of 10. This representation significantly simplifies the handling of very large or very small numbers, making them more manageable in calculations and comparisons. For instance, the number 3,000,000,000 can be expressed in scientific notation as 3 × 10⁹, and the number 0.000000005 can be written as 5 × 10⁻⁹. The exponent of 10 indicates the number of places the decimal point must be moved to obtain the original number. A positive exponent signifies a large number, while a negative exponent indicates a small number.
Converting to Scientific Notation
Before we proceed with the multiplication, it's crucial to express both numbers in scientific notation. The first number, 0.0000003, needs to be converted. To do this, we move the decimal point seven places to the right, resulting in 3. The exponent of 10 will be -7, as we moved the decimal point to the right (indicating a number smaller than 1). Therefore, 0.0000003 in scientific notation is 3 × 10⁻⁷. The second number, 9.1 × 10⁻¹⁵, is already in scientific notation.
The Multiplication Process
Now that both numbers are in scientific notation, we can perform the multiplication: (3 × 10⁻⁷) × (9.1 × 10⁻¹⁵). The multiplication process involves two key steps:
- Multiply the coefficients: Multiply the decimal parts of the numbers (3 and 9.1). 3 multiplied by 9.1 equals 27.3.
- Multiply the powers of 10: Multiply the exponential parts of the numbers (10⁻⁷ and 10⁻¹⁵). According to the rules of exponents, when multiplying powers with the same base, we add the exponents. Thus, 10⁻⁷ × 10⁻¹⁵ = 10^(⁻⁷ + ⁻¹⁵) = 10⁻²². The intermediate result of the multiplication is 27.3 × 10⁻²².
Expressing the Result in Standard Form
While 27.3 × 10⁻²² is a mathematically correct result, it's not in standard scientific notation because 27.3 is not a number between 1 and 10. To convert it to standard form, we need to move the decimal point one place to the left, making the coefficient 2.73. Moving the decimal point to the left increases the exponent by 1. Therefore, we adjust the exponent of 10 by adding 1 to -22, resulting in -21. The final answer in standard form is 2.73 × 10⁻²¹.
In summary, we converted 0.0000003 to 3 × 10⁻⁷, multiplied it by 9.1 × 10⁻¹⁵ to get 27.3 × 10⁻²², and then converted the result to standard scientific notation, which is 2.73 × 10⁻²¹. This methodical approach ensures accuracy and clarity when dealing with numbers in scientific notation.
Detailed Solution: Step-by-Step Breakdown
To further solidify your understanding, let's dissect the solution into a detailed, step-by-step breakdown. This approach not only clarifies the process but also highlights the underlying principles that govern operations with scientific notation. By carefully examining each step, you'll gain a deeper appreciation for the elegance and efficiency of this mathematical tool.
Step 1: Convert 0.0000003 to Scientific Notation
The initial number, 0.0000003, is a decimal less than 1. To convert it to scientific notation, we need to express it as a number between 1 and 10 multiplied by a power of 10. The process involves moving the decimal point to the right until we obtain a number within the desired range. In this case, we move the decimal point seven places to the right:
- 0000003 → 0.000003 → 0.00003 → 0.0003 → 0.003 → 0.03 → 0.3 → 3
This transformation gives us the number 3. Since we moved the decimal point seven places to the right, the exponent of 10 will be -7. This is because we are dealing with a number smaller than 1, and each movement of the decimal point to the right corresponds to a decrease in the exponent. Therefore, 0.0000003 in scientific notation is 3 × 10⁻⁷.
Step 2: Identify the Second Number in Scientific Notation
The second number provided, 9.1 × 10⁻¹⁵, is already expressed in scientific notation. This simplifies our task, as we can directly proceed with the multiplication. It's important to recognize that 9.1 is a number between 1 and 10, and 10⁻¹⁵ represents the power of 10. This number is exceedingly small, indicating that the decimal point in 9.1 would need to be moved 15 places to the left to obtain its standard decimal form.
Step 3: Multiply the Numbers in Scientific Notation
Now that both numbers are in scientific notation, we can perform the multiplication:
(3 × 10⁻⁷) × (9.1 × 10⁻¹⁵)
To multiply numbers in scientific notation, we follow two main rules:
- Multiply the coefficients (the numbers between 1 and 10).
- Multiply the powers of 10 by adding their exponents.
Applying these rules:
- Multiply the coefficients: 3 × 9.1 = 27.3
- Multiply the powers of 10: 10⁻⁷ × 10⁻¹⁵ = 10^(⁻⁷ + ⁻¹⁵) = 10⁻²²
Combining these results, we get an intermediate answer of 27.3 × 10⁻²².
Step 4: Convert the Result to Standard Scientific Notation
The result 27.3 × 10⁻²² is not yet in standard scientific notation because 27.3 is greater than 10. To convert it to standard form, we need to adjust the decimal point so that the coefficient is a number between 1 and 10. We move the decimal point one place to the left:
- 3 → 2.73
Moving the decimal point one place to the left increases the exponent of 10 by 1. Therefore, we add 1 to the current exponent, -22:
- 22 + 1 = -21
Thus, the final result in standard scientific notation is 2.73 × 10⁻²¹.
By breaking down the solution into these steps, we've not only solved the problem but also reinforced the methodology behind manipulating numbers in scientific notation. This structured approach is crucial for handling complex calculations and ensuring accuracy in mathematical problem-solving.
Common Mistakes and How to Avoid Them
When working with scientific notation, several common mistakes can occur. Being aware of these pitfalls and understanding how to avoid them is crucial for achieving accurate results. Let's examine some frequent errors and the strategies to prevent them.
Misunderstanding Scientific Notation
One of the primary errors is a misunderstanding of the fundamental concept of scientific notation. Remember, scientific notation expresses a number as a product of a coefficient (a number between 1 and 10) and a power of 10. Failing to adhere to this format can lead to incorrect conversions and calculations.
How to avoid it: Always ensure that the coefficient is between 1 and 10. If it's not, adjust the decimal point and the exponent accordingly. For example, if you have 35 × 10⁻⁶, it's not in scientific notation. You need to rewrite it as 3.5 × 10⁻⁵ by moving the decimal one place to the left and increasing the exponent by 1.
Incorrect Decimal Point Movement
Moving the decimal point in the wrong direction or by the wrong number of places is another common mistake. This can happen when converting numbers to or from scientific notation.
How to avoid it: When converting a number to scientific notation, carefully count how many places you move the decimal point. If you move it to the left, the exponent will be positive (for large numbers). If you move it to the right, the exponent will be negative (for small numbers). For example, converting 5,000,000 requires moving the decimal point six places to the left, resulting in 5 × 10⁶. Conversely, converting 0.00004 involves moving the decimal point five places to the right, yielding 4 × 10⁻⁵.
Errors in Exponent Arithmetic
When multiplying or dividing numbers in scientific notation, mistakes can arise in handling the exponents. Remember the rules of exponents: when multiplying, add the exponents; when dividing, subtract the exponents.
How to avoid it: Double-check your exponent arithmetic. For instance, if you are multiplying (2 × 10⁴) by (3 × 10⁵), you add the exponents: 4 + 5 = 9. So, the result is 6 × 10⁹. Similarly, when dividing (6 × 10⁷) by (2 × 10³), you subtract the exponents: 7 - 3 = 4, resulting in 3 × 10⁴. Pay close attention to negative exponents; adding or subtracting them requires careful consideration of the signs.
Forgetting to Adjust After Multiplication or Division
After performing multiplication or division, the resulting coefficient might not be in the required range of 1 to 10. This necessitates an additional adjustment to convert the answer into proper scientific notation.
How to avoid it: Always review your final answer. If the coefficient is not between 1 and 10, move the decimal point accordingly and adjust the exponent. For example, if your calculation yields 45 × 10⁻³, you need to move the decimal point one place to the left, making it 4.5, and increase the exponent by 1, resulting in 4.5 × 10⁻².
Calculator Errors
Calculators can be incredibly helpful, but they can also be a source of errors if not used correctly. Entering numbers in scientific notation requires using the appropriate exponent keys (often labeled as EXP or EE), and misusing these keys can lead to incorrect results.
How to avoid it: Familiarize yourself with your calculator's functions for scientific notation. Practice entering numbers and performing calculations to ensure you understand how to use the exponent keys correctly. Always double-check the display to confirm that the numbers are entered accurately. It's also wise to estimate the answer mentally to catch any significant discrepancies.
By understanding these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy and confidence when working with scientific notation. Consistent practice and attention to detail are key to mastering this essential mathematical skill.
Real-World Applications of Scientific Notation
Scientific notation is not just a mathematical abstraction; it's an indispensable tool in various scientific and technical fields. Its ability to express extremely large and small numbers concisely makes it crucial for calculations and representations across numerous disciplines. Let's explore some real-world applications of scientific notation to appreciate its practical significance.
Astronomy
In astronomy, the distances between celestial bodies are vast, and their masses are enormous. Scientific notation is essential for expressing these quantities in a manageable format. For instance, the distance to the nearest star, Proxima Centauri, is approximately 4.246 light-years, which translates to about 4.017 × 10¹³ kilometers. The mass of the Sun is approximately 1.989 × 10³⁰ kilograms. These numbers would be cumbersome and prone to errors if written in their full decimal form. Scientific notation allows astronomers to perform calculations, compare values, and communicate findings effectively.
Physics
Physics deals with a wide range of scales, from the subatomic to the cosmic. Physicists often encounter extremely small values, such as the mass of an electron (approximately 9.109 × 10⁻³¹ kilograms) and extremely large values, such as the speed of light (approximately 2.998 × 10⁸ meters per second). Scientific notation is vital for expressing these values accurately and performing calculations in areas such as quantum mechanics, electromagnetism, and cosmology. It simplifies the manipulation of physical constants and variables, making complex equations more tractable.
Chemistry
In chemistry, the scale of atoms and molecules necessitates the use of very small numbers. Avogadro's number, approximately 6.022 × 10²³, is a fundamental constant that represents the number of atoms or molecules in a mole of a substance. The masses of individual atoms and molecules are also exceedingly small and are often expressed in atomic mass units (amu), where 1 amu is approximately 1.661 × 10⁻²⁷ kilograms. Scientific notation is essential for stoichiometric calculations, expressing concentrations, and understanding chemical reactions at the molecular level.
Computer Science
Computer science deals with data storage, processing speeds, and network capacities that can range from very small to very large values. For example, a nanosecond (one billionth of a second) is 1 × 10⁻⁹ seconds, and a terabyte of data is approximately 1 × 10¹² bytes. Scientific notation is used to describe these quantities and to analyze the performance of algorithms, the capacity of storage devices, and the speed of data transmission. It helps in understanding the scale of computational operations and the efficiency of computer systems.
Engineering
Various engineering disciplines rely on scientific notation to handle calculations involving large structures, tiny components, and complex systems. Civil engineers may use it to calculate stresses and strains in buildings and bridges, while electrical engineers use it to analyze circuits and electromagnetic fields. Chemical engineers employ scientific notation in process design and optimization, and mechanical engineers use it in thermodynamics and fluid mechanics. The ability to express and manipulate large and small numbers accurately is crucial for design, analysis, and problem-solving in engineering.
Medicine and Biology
In medicine and biology, scientific notation is used to represent concentrations of substances in the body, sizes of microorganisms, and the scale of biological processes. The diameter of a typical bacterium might be around 1 × 10⁻⁶ meters, and the concentration of a drug in the bloodstream might be expressed in nanomoles per liter (e.g., 5 × 10⁻⁹ moles/L). Scientific notation is essential for understanding physiological processes, diagnosing diseases, and developing treatments.
These examples illustrate that scientific notation is a versatile and essential tool in a wide range of disciplines. Its ability to simplify the representation and manipulation of extremely large and small numbers makes it indispensable for scientific research, technological innovation, and practical applications across various fields.
Conclusion
In conclusion, the problem 0.0000003 × (9.1 × 10⁻¹⁵) exemplifies the importance of scientific notation in handling very small numbers. By converting 0.0000003 to scientific notation (3 × 10⁻⁷), multiplying the coefficients (3 and 9.1) and the powers of 10 (10⁻⁷ and 10⁻¹⁵), and then adjusting the result to standard form, we arrived at the solution 2.73 × 10⁻²¹. This process underscores the significance of understanding scientific notation and its applications. Scientific notation simplifies complex calculations, enhances clarity in numerical representations, and is crucial across numerous scientific and technical disciplines. Mastering this skill empowers individuals to tackle a wide range of mathematical challenges with confidence and precision, making it an invaluable asset in both academic and professional pursuits.
