Question

What is the sum of \(5.786 \cdot 10^{3} \mathrm{~m}\) and \(3.19 \cdot 10^{4} \mathrm{~m} ?\) a) \(6.02 \cdot 10^{23} \mathrm{~m}\) c) \(8.976 \cdot 10^{3} \mathrm{~m}\) b) \(3.77 \cdot 10^{4} \mathrm{~m}\) d) \(8.98 \cdot 10^{3} \mathrm{~m}\)

Short answer

Answer: The sum of \(5.786 \cdot 10^3 \mathrm{~m}\) and \(3.19 \cdot 10^4 \mathrm{~m}\) in scientific notation is \(3.77 \cdot 10^4 \mathrm{~m}\).

Step-by-step solution

Arrange the numbers to be added

First, write the given numbers in a columnar format, aligning their decimal points: \(5.786 \cdot 10^3 \mathrm{~m}\) \(+ \ 3.19 \cdot 10^4 \mathrm{~m}\)

Match the exponents

To add the numbers, the exponent part should be the same. You will need to adjust the smaller exponent number: \(57.86 \cdot 10^2 \mathrm{~m}\) \(+ \ 3.19 \cdot 10^4 \mathrm{~m}\) Now, both numbers have a \(10^2\) base: \(57.86 \cdot 10^2 \mathrm{~m}\) \(+ \ 319 \cdot 10^2 \mathrm{~m}\)

Add the numbers

Now that both numbers have the same exponent, you can add them: \((57.86 + 319) \cdot 10^2 \mathrm{~m}\) \(= 376.86 \cdot 10^2 \mathrm{~m}\)

Convert back to scientific notation

Convert the sum back to scientific notation: \(3.7686 \cdot 10^4 \mathrm{~m}\)

Round off if necessary

Now, round off the answer to the given significant figures: \(3.77 \cdot 10^4 \mathrm{~m}\)

Check the answer in the options

Compare your answer to the given options. Your answer matches option b: b) \(3.77 \cdot 10^4 \mathrm{~m}\)

Key concepts

Adding Scientific Notation

When combining measurements or values expressed in scientific notation, it's crucial to ensure they share the same exponent for accurate addition. Let's explore the fundamentals:
Scientific notation represents numbers as a product of two parts: a decimal (between 1 and 10) and a power of ten. For example, in the expression \(5.786 \cdot 10^{3}\), the decimal is 5.786, and the exponent is 3.

• To add numbers in scientific notation, first modify the exponents so they match. This usually involves converting the smaller exponent to match the larger one.
• Adjust the decimal part accordingly, so the scientific notations align properly.
• Add the aligned decimal parts, then multiply the result by the common power of ten.
• If necessary, adjust the result back into standard scientific notation format.

Using the method ensures precision, allowing measurements from diverse scales to be combined consistently.

Significant Figures in Physics

Significant figures play a pivotal role in physics, shaping how accurately we report measurements and calculations. Each digit in a measurement is deemed 'significant' if it contributes precise information.

• The number of significant figures reflects the precision of a measurement.
• Rules for identifying significant figures include recognizing all non-zero digits as significant and considering zeroes based on their position.
• When adding or subtracting, the answer should not have more decimal places than the least precise measurement.
• In multiplication and division, the number of significant figures in the result is dictated by the original number with the fewest significant figures.

Maintaining the correct number of significant figures ensures the legitimacy and comparability of scientific findings.

Exponent Manipulation

Mastering exponent manipulation is essential for handling scientific notation and algebraic expressions efficiently. Exponent rules allow us to multiply, divide, and take powers or roots of numbers with exponents without expanding them completely.

• The 'Power Rule' indicates that when raising a power to another power, you multiply the exponents.
• The 'Quotient Rule' specifies that when dividing powers with the same base, you subtract the exponents.
• The 'Product Rule' implies that when multiplying powers with the same base, you add the exponents.

Understanding and applying these precepts allows us to seamlessly adjust the exponents in scientific notation during operations such as addition, which is vital for maintaining correct formatting and ensuring precision in scientific and mathematical calculations.