Question

Mass of earth is $5.97\times {{10}^{24}}$ kg and the mass of the moon is $7.35\times {{10}^{22}}$ kg. What is the total mass?

Answer 1

Hint: First of all make the exponent of base 10 of both the masses of earth and moon equal to each other. To do this, use the formula of exponents ${{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}$ and write ${{10}^{24}}={{10}^{2}}\times {{10}^{22}}$ in the mass of earth. Now, multiply ${{10}^{2}}$ with 5.97 and add the two masses by taking ${{10}^{22}}$ common from them. Add the decimal numbers to get the answer.

Complete step by step answer:
Here we have been provided with the mass of earth equal to $5.97\times {{10}^{24}}$ kg and the mass of moon equal to $7.35\times {{10}^{22}}$ kg. We have been asked to find the total mass, i.e. we need to add the two masses.
Now, the masses contain exponents of 10 and the exponents are different. So to add them we need to make the exponents of 10 in the two masses equal to each other. Let us make the exponent of 10 in the mass of earth equal to 22 since the exponent of 10 in the mass of the moon is also 22.
$\Rightarrow $ Mass of earth = $5.97\times {{10}^{24}}$ kg
We can write the exponent as ${{10}^{24}}={{10}^{\left( 22+2 \right)}}$, using the formula of exponent given as ${{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}$ we get,
$\Rightarrow $ Mass of earth = $5.97\times {{10}^{2}}\times {{10}^{22}}$ kg
$\Rightarrow $ Mass of earth = $597\times {{10}^{22}}$ kg
Adding the masses of the moon and the earth we get,
$\Rightarrow $ Total mass = $\left[ \left( 597\times {{10}^{22}} \right)+\left( 7.35\times {{10}^{22}} \right) \right]$ kg
$\Rightarrow $ Total mass = $\left( 597+7.35 \right)\times {{10}^{22}}$ kg
$\Rightarrow $ Total mass = $604.35\times {{10}^{22}}$ kg
 Hence, the total mass of the two objects is $604.35\times {{10}^{22}}$=$6.04\times {{10}^{24}}$ kg.

Note: Note that you can also write the answer in which the exponent of 10 will be 24. What you have to do is multiply the obtained answer with ${{10}^{2}}$ and to balance it divide the expression with the same number. Use the formula of exponent ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$ to simplify the exponential term and to simplify the decimal number take the decimal point two places to the left.