Question

The Earth's mass is \(5.9742 \times 10^{24}\) kilograms, while the moon's mass is \(7.36 \times 10^{22}\) kilograms. What is an estimate of the sum of the masses, expressed in scientific notation? A. \(1.33 \times 10^{24}\) B. \(1.39 \times 10^{24}\) C. \(6.05 \times 10^{24}\) D. \(1.33 \times 10^{24}\)

Short answer

The sum is approximately \(6.05 \times 10^{24} \) kg, which matches option C.

Step-by-step solution

Identify the masses and their exponents

The Earth's mass is given as \(\text 5.9742 \times 10^{24} \) kg and the Moon's mass is \(\text 7.36 \times 10^{22} \) kg. We need to estimate the sum using scientific notation.

Adjust the powers of 10

Convert the Moon's mass to a compatible power of 10. \(\text 7.36 \times 10^{22} = 0.0736 \times 10^{24} \).

Add the coefficients

Now, add the coefficients for the two masses: \(\text 5.9742 \ + \text 0.0736 = 6.0478 \).

Simplify the sum

Round \(\text 6.0478 \) to a reasonable precision. Approximately, \(\text 6.05 \). Therefore, the sum is \(6.05 \times 10^{24} \) kg.

Key concepts

Mass Estimation

Estimating mass, especially in astronomy, involves dealing with very large numbers, often written using scientific notation. Scientific notation helps manage and simplify these large values. When estimating mass, ensure you're aware of rounding rules to keep precision manageable without losing significant value. This is crucial when performing operations like addition or subtraction, which could introduce small errors if not managed correctly. Here, the Earth's mass and the Moon's mass are given in scientific notation and need to be summed up. This requires understanding both the coefficients and the exponents involved.

Powers of 10

Powers of 10 are used to express very large or very small numbers conveniently. They consist of a coefficient (a number) and an exponent indicating how many times to multiply the coefficient by 10. For example, in scientific notation, the Earth's mass is written as \(5.9742 \times 10^{24} \) kilograms. Here, 5.9742 is the coefficient and 24 is the exponent. When dealing with powers of 10, if we need to perform operations like addition or subtraction, it's essential first to convert all numbers to the same power of ten. For the Moon's mass \(7.36 \times 10^{22} \), it is converted to \(0.0736 \times 10^{24} \) to match the Earth's format allowing easy addition.

Summation in Scientific Notation

Summation in scientific notation involves aligning the exponents and then performing the arithmetic on the coefficients. After aligning the exponents, you add or subtract the coefficients like regular numbers. In our problem, after converting the Moon's mass to \(0.0736 \times 10^{24} \), we add it to the Earth's mass coefficient: \(5.9742 + 0.0736 = 6.0478 \). We then round this result to a reasonable precision. Here, it’s simplified to approximately \(6.05 \). Finally, we combine this new coefficient with the common exponent, giving us the sum \(6.05 \times 10^{24} \) kilograms. This method ensures that operations on large numbers remain precise and manageable.