Problem 20 - Basic Engineering Mathematics

Question

Find the value of (a) \(7.9 \times 10^{-2}-5.4 \times 10^{-2}\) (b) \(8.3 \times 10^{3}+5.415 \times 10^{3}\) and (c) \(9.293 \times 10^{2}+1.3 \times 10^{3}\) expressing the answers in standard form.

Step-by-Step Solution

Answer

(a) \(2.5 \times 10^{-2}\); (b) \(1.3715 \times 10^{4}\); (c) \(2.2293 \times 10^{3}\).

Step 1: Subtract Small Numbers

For part (a), subtract the two numbers in scientific notation: \(7.9 \times 10^{-2} - 5.4 \times 10^{-2}\). Combine the numerical coefficients: \(7.9 - 5.4 = 2.5\). Thus, the result is: \(2.5 \times 10^{-2}\).

Step 2: Add Large Numbers

For part (b), add the numbers in scientific notation: \(8.3 \times 10^{3} + 5.415 \times 10^{3}\). Since the power of 10 is the same, add the coefficients:\(8.3 + 5.415 = 13.715\). The result is: \(1.3715 \times 10^{4}\) (after adjusting to put it in standard form by moving the decimal one place to the left and increasing the power by one).

Step 3: Add Different Powers of Ten

For part (c), add the numbers: \(9.293 \times 10^{2} + 1.3 \times 10^{3}\). Convert \(1.3 \times 10^{3}\) to \(13.0 \times 10^{2}\) to match powers of ten, then add the coefficients:\(9.293 + 13.0 = 22.293\). Hence, the result is: \(2.2293 \times 10^{3}\).

Key Concepts

Standard Form

Scientific notation is a way to express very large or very small numbers conveniently. In this system, numbers are represented as a product of a number (usually between 1 and 10) and a power of ten. The standard form simplifies both the writing and reading of such numbers, making them much more manageable.
The standard form of a number is written as:

• "Coefficient" × "Power of Ten"
• The coefficient should be between 1 and 10 (excluding 10).
• The power of ten indicates how many places to move the decimal point.

For instance, the number 5,670 can be expressed in the standard form as:
\[5.67 \times 10^{3}\]This is adjusted from 56.7 by moving the decimal two places to the right and adjusting the exponent from zero to three. Using standard form helps in performing calculations more efficiently, especially with multiplication or division.

Numerical Coefficients

Numerical coefficients are the numbers that multiply the power of ten in scientific notation. In the example of scientific notation, such as 3.5 × 10⁴, 3.5 is the numerical coefficient.
They are crucial during mathematical operations like addition or subtraction within standard form. Here's why:

• When adding or subtracting in scientific notation, the coefficients are directly operated upon.
• The power of ten remains unchanged if they are the same for all terms involved.
• Changes in the coefficients can influence how exponents are adjusted, especially if they result in a coefficient that requires normalization (adapting so it's between 1 and 10).

For example, in the case of addition \(8.3 \times 10^3 + 5.415 \times 10^3\), coefficients are added directly to result in \(13.715 \), creating \(1.3715 \times 10^4\) when converted to standard form. The initial sum of the coefficients may need adjustment to remain in the correct form.

Powers of Ten

Powers of ten indicate the number of times 10 is multiplied by itself, providing a compact way to express large numbers or small fractions. In scientific notation, powers of ten play a significant role by anchoring the positioning of the decimal point.
When numbers are expressed as powers of ten, it helps in determining how far and in which direction the decimal point must move:

• Positive exponents signify that the decimal point moves to the right, scaling up the value.
• Negative exponents mean the decimal point shifts left, making the number smaller.

For example, in the calculation of \(2.5 \times 10^{-2}\) from \(7.9 \times 10^{-2} - 5.4 \times 10^{-2}\), the power \(-2\) keeps the result in a fractional form because the decimal moves left.
Computation involving different powers requires adaptation, as seen in \(9.293 \times 10^{2} + 1.3 \times 10^{3}\). Here, it's crucial to adjust the numbers to a common power (preferring smaller exponents) for straightforward operations, before normalization of the results.