Question

Complete the following addition and subtraction problems in scientific notation. a. \(\left(6.23 \times 10^{6} \mathrm{kL}\right)+\left(5.34 \times 10^{6} \mathrm{kL}\right)\) b. \(\left(3.1 \times 10^{4} \mathrm{mm}\right)+\left(4.87 \times 10^{5} \mathrm{mm}\right)\) c. \(\left(7.21 \times 10^{3} \mathrm{mg}\right)+\left(43.8 \times 10^{2} \mathrm{mg}\right)\) d. \(\left(9.15 \times 10^{-4} \mathrm{cm}\right)+\left(3.48 \times 10^{-4} \mathrm{cm}\right)\) e. \(\left(4.68 \times 10^{-5} \mathrm{cg}\right)+\left(3.5 \times 10^{-6} \mathrm{cg}\right)\) f. \(\left(3.57 \times 10^{2} \mathrm{mL}\right)-\left(1.43 \times 10^{2} \mathrm{mL}\right)\) g. \(\left(9.87 \times 10^{4} \mathrm{g}\right)-\left(6.2 \times 10^{3} \mathrm{g}\right)\) h. \(\left(7.52 \times 10^{5} \mathrm{kg}\right)-\left(5.43 \times 10^{5} \mathrm{kg}\right)\) i. \(\left(6.48 \times 10^{-3} \mathrm{mm}\right)-\left(2.81 \times 10^{-3} \mathrm{mm}\right)\) j. \(\left(5.72 \times 10^{-4} \mathrm{dg}\right)-\left(2.3 \times 10^{-5} \mathrm{dg}\right)\)

Short answer

The short answers are: a. \(11.57 \times 10^{6} \mathrm{kL}\) b. \(51.8 \times 10^4 \mathrm{mm}\) c. \(11.59 \times 10^3 \mathrm{mg}\) d. \(12.63 \times 10^{-4} \mathrm{cm}\) e. \(5.03 \times 10^{-5} \mathrm{cg}\) f. \(2.14 \times 10^2 \mathrm{mL}\) g. \(9.25 \times 10^4 \mathrm{g}\) h. \(2.09 \times 10^5\mathrm{kg}\) i. \(3.67 \times 10^{-3} \mathrm{mm}\) j. \(5.49 \times 10^{-4}\mathrm{dg}\)

Step-by-step solution

a. Addition of given numbers in scientific notation

To add two numbers in scientific notation, make sure the exponents are the same. In this case, the exponents are the same: \(10^6\). \( (6.23 \times 10^{6} \mathrm{kL}) + (5.34 \times 10^{6} \mathrm{kL}) = (6.23 + 5.34) \times 10^{6} \mathrm{kL} = 11.57 \times 10^{6} \mathrm{kL} \)

b. Addition of given numbers in scientific notation

First, rewrite the numbers with the same exponent. \( (3.1 \times 10^4 \mathrm{mm}) + (4.87 \times 10^5 \mathrm{mm}) = (3.1 \times 10^4 \mathrm{mm}) + (48.7 \times 10^4 \mathrm{mm}) \) Now, add the numbers. \( (3.1 + 48.7) \times 10^4 \mathrm{mm} = 51.8 \times 10^4 \mathrm{mm} \)

c. Addition of given numbers in scientific notation

First, rewrite the numbers with the same exponent. \( (7.21 \times 10^3 \mathrm{mg}) + (43.8 \times 10^2 \mathrm{mg}) = (7.21 \times 10^3 \mathrm{mg}) + (4.38 \times 10^3 \mathrm{mg}) \) Now, add the numbers. \( (7.21 + 4.38) \times 10^3 \mathrm{mg} = 11.59 \times 10^3 \mathrm{mg} \)

d. Addition of given numbers in scientific notation

In this case, the exponents are the same: \(10^{-4}\). \( (9.15 \times 10^{-4} \mathrm{cm}) + (3.48 \times 10^{-4} \mathrm{cm}) = (9.15 + 3.48) \times 10^{-4} \mathrm{cm} = 12.63 \times 10^{-4} \mathrm{cm} \)

e. Addition of given numbers in scientific notation

First, rewrite the numbers with the same exponent. \( (4.68 \times 10^{-5} \mathrm{cg}) + (3.5 \times 10^{-6} \mathrm{cg}) = (4.68 \times 10^{-5} \mathrm{cg}) + (0.35 \times 10^{-4} \mathrm{cg}) \) Now, add the numbers. \( (4.68 + 0.35) \times 10^{-5} \mathrm{cg} = 5.03 \times 10^{-5} \mathrm{cg} \)

f. Subtraction of given numbers in scientific notation

In this case, the exponents are the same: \(10^2\). \( (3.57 \times 10^2 \mathrm{mL}) - (1.43 \times 10^2 \mathrm{mL}) = (3.57 - 1.43) \times 10^2 \mathrm{mL} = 2.14 \times 10^2 \mathrm{mL} \)

g. Subtraction of given numbers in scientific notation

First, rewrite the numbers with the same exponent. \( (9.87 \times 10^4 \mathrm{g}) - (6.2 \times 10^3 \mathrm{g}) = (9.87 \times 10^4 \mathrm{g}) - (0.62 \times 10^4 \mathrm{g}) \) Now, subtract the numbers. \( (9.87 - 0.62) \times 10^4 \mathrm{g} = 9.25 \times 10^4 \mathrm{g} \)

h. Subtraction of given numbers in scientific notation

In this case, the exponents are the same: \(10^5\). \( (7.52 \times 10^5\mathrm{kg}) - (5.43 \times 10^5\mathrm{kg}) = (7.52 - 5.43) \times 10^5\mathrm{kg} = 2.09 \times 10^5\mathrm{kg} \)

i. Subtraction of given numbers in scientific notation

In this case, the exponents are the same: \(10^{-3}\). \( (6.48 \times 10^{-3} \mathrm{mm}) - (2.81 \times 10^{-3} \mathrm{mm}) = (6.48 - 2.81) \times 10^{-3} \mathrm{mm} = 3.67 \times 10^{-3} \mathrm{mm} \)

j. Subtraction of given numbers in scientific notation

First, rewrite the numbers with the same exponent. \( (5.72 \times 10^{-4}\mathrm{dg}) - (2.3 \times 10^{-5}\mathrm{dg}) = (5.72 \times 10^{-4}\mathrm{dg}) - (0.23 \times 10^{-3}\mathrm{dg}) \) Now, subtract the numbers. \( (5.72 - 0.23) \times 10^{-4}\mathrm{dg} = 5.49 \times 10^{-4}\mathrm{dg} \)

Key concepts

Exponent Rules

Understanding exponent rules is essential when working with scientific notation. These rules clarify how to handle numbers involving powers of ten. Here are some basic rules:

• Multiplication: When multiplying numbers with the same base, you add their exponents. For instance, \(10^a \times 10^b = 10^{a+b}\).
• Division: When dividing, you subtract the exponents. \(10^a / 10^b = 10^{a-b}\).
• Power of a Power: Multiply the exponents when raising a power to another power: \((10^a)^b = 10^{a \cdot b}\).
• Zero Exponent: Any number raised to the power of zero is 1: \(10^0 = 1\).

These rules make it easier to simplify expressions and perform operations needed in scientific notation.

Addition and Subtraction

When adding or subtracting in scientific notation, ensure the exponents match before performing any calculations. This allows the coefficients to combine directly:

• If the exponents are the same, simply add or subtract the coefficients.
• If the exponents differ, you must adjust one number so both have the same exponent. Choose to adjust the smaller exponent up to prevent handling very small decimal numbers in the coefficient.

Once the exponents are equal, treat the coefficients like regular numbers. This method maintains accuracy and clarity while dealing with very large or very small values.

Problem-Solving Techniques

To effectively solve problems involving scientific notation, follow these steps:

• Analyze the problem: Identify whether it involves addition, subtraction, multiplication, or division.
• Standardize exponents: Make sure each term has the same exponent if addition or subtraction is needed. This usually involves increasing or decreasing one of the exponents so they match.
• Perform arithmetic: Add or subtract coefficients once exponents are equal. For multiplication or division, directly apply the exponent rules.
• Simplify: Ensure the final result is in standard scientific notation, with only one non-zero digit to the left of the decimal point.

By following this structured approach, you can confidently tackle scientific notation problems.

Mathematical Notation

Mathematical notation provides a universal language for expressing numbers and operations clearly and concisely. Scientific notation is part of this language, especially useful for representing extremely large or small numbers.

• Format: It takes the form of \(a \times 10^b\), where \(1 \leq a < 10\) and \(b\) is an integer representing the number of decimal places moved.
• Purpose: This notation saves space and simplifies calculations, especially helpful in fields like physics, chemistry, and engineering.
• Consistency: By using standard notation rules, mathematical expressions remain consistent across various fields and publications, aiding global communication of ideas.

Scientific notation is a practical tool in both academic and professional settings, helping to streamline complex calculations and communicate effectively.