Problem 62 - Technical Mathematics with Calculus

Question

Perform the following computations. Display your answer in scientific notation. $$\left(2.74 \times 10^{3}\right) \div\left(9.13 \times 10^{5}\right)$$

Step-by-Step Solution

Answer

\(3.00 \times 10^{-1}\)

Step 1: Divide the coefficients

Divide the numerical parts (coefficients) of the given numbers, ignoring the powers of 10 for now. In this case, divide 2.74 by 9.13.

Step 2: Simplify the division

Perform the division to find the quotient of the coefficients. The result is approximately 0.300.

Step 3: Subtract the exponents

When dividing exponential numbers with the same base, subtract the exponent of the divisor from the exponent of the dividend. In this case, subtract 5 from 3, which is \(3-5=-2\).

Step 4: Convert to scientific notation

Combine the result from Step 2 with the result from Step 3 to write the answer in scientific notation. However, because scientific notation requires the first number to be greater than or equal to 1 and less than 10, we need to adjust the coefficient and exponent. Move the decimal point one place to the right to get 3.00 and add one to the exponent to get -1.

Key Concepts

Dividing Exponential Numbers

Dividing exponential numbers often appears daunting, but with a strong grasp of the basic rules, it's a straightforward process. When faced with exponential numbers with the same base being divided, one must only subtract the exponent of the divisor from the exponent of the dividend. To digest this, let's focus on an example: Imagine you have the problem \( 10^3 \div 10^5 \). Rather than performing long division, you can simplify your work by subtracting the exponents directly. Here, you would compute \( 3 - 5 = -2 \), resulting in \( 10^{-2} \). When working through such problems:

• Keep the base constant unless instructed otherwise,
• Remember to align your terms so that you can easily identify the exponents to subtract,
• Practice problems at various difficulty levels to strengthen understanding.

Utilize these steps to master dividing numbers with exponential terms and enhance your proficiency in the mathematical concept.

Scientific Notation in Mathematics

Scientific notation is a convenient way to express very large or very small numbers, which are frequently encountered in scientific and mathematical calculations. The format typically involves a coefficient between 1 and 10 and a power of 10.In scientific notation, a number like \( 48000000 \) is written as \( 4.8 \times 10^7 \) which keeps the notation compact and error-free when dealing with extreme values. Key aspects to understanding scientific notation include:

• Identifying the coefficient, which is a single non-zero digit or a decimal followed by non-zero digits,
• Recognizing the base, which is always 10 in scientific notation,
• The exponent, which indicates the number of places the decimal point must move to convert the number back to its original form.

By becoming familiar with scientific notation, not only can you accurately work with large data sets, but also ensure precision in your mathematical expressions.

Computing Coefficients

In the context of scientific notation and algebra, coefficients play a pivotal role. They are the numerical part of terms, which are front and center when performing operations like addition, subtraction, multiplication, and division. It is particularly important to pay attention to coefficients when dividing numbers in scientific notation. For instance, if you have the expression \( 2.74 \times 10^3 \div 9.13 \times 10^5 \), your initial step would be to divide the coefficients: \( 2.74 \div 9.13 = 0.300 \).While computing coefficients:

• Perform the operation as you would with any decimal numbers,
• Ensure that your result is expressed with the appropriate level of precision,
• Consider the context – does your coefficient need to be adjusted to fit back into proper scientific notation form?

The mastery of computing coefficients anchors your fundamental understanding of scientific notation and enhances your ability to handle variables and constants across various mathematical concepts.