Question

solve. write the answer using proper scientific notation.
1
the earth is 9.296 x 10⁶ miles from the sun. mars is 1.416 x 10⁷ miles from the sun. how much farther is earth from the sun than mars?
2
the mass of the sun is 1.989 x 10³⁰ kg. the mass of the earth is 5.972 x 10²⁴ kg. what is the difference in the mass of the sun and earth?

Explanation:

Problem 1

Step1: Identify the distances

Earth's distance: $9.296 \times 10^{7}$ miles (assuming a typo, as $9.296\times10^6$ would be closer than Mars, likely $10^7$). Mars' distance: $1.416 \times 10^{7}$ miles.

Step2: Subtract the distances

Difference = $9.296\times 10^{7}- 1.416\times 10^{7}$
Factor out $10^{7}$: $(9.296 - 1.416)\times10^{7}$
Calculate $9.296 - 1.416 = 7.88$
So difference is $7.88\times 10^{7}$ miles. (If original Earth distance was $9.296\times10^6$, then convert Mars to $14.16\times10^6$, difference $9.296 - 14.16$ is negative, so correct Earth distance should be $9.296\times10^7$)

Step1: Identify the masses

Sun's mass: $1.989\times 10^{30}$ kg. Earth's mass: $5.972\times 10^{24}$ kg.

Step2: Convert to same exponent

Convert Earth's mass to $0.000005972\times 10^{30}$ kg (since $10^{24}=10^{-6}\times10^{30}$)

Step3: Subtract the masses

Difference = $1.989\times 10^{30}- 0.000005972\times 10^{30}$
Factor out $10^{30}$: $(1.989 - 0.000005972)\times10^{30}$
Calculate $1.989 - 0.000005972\approx1.988994028$
So difference is approximately $1.988994028\times 10^{30}$ kg, or more neatly $1.989\times 10^{30}- 5.972\times 10^{24}=1.989\times 10^{30}- 0.000005972\times 10^{30}=(1.989 - 0.000005972)\times 10^{30}\approx1.989\times 10^{30}$ kg (since $5.972\times 10^{24}$ is much smaller than $1.989\times 10^{30}$)

Answer:

$7.88\times 10^{7}$ miles

Problem 2