Question

The volume of the earth is approximately $7.67\times {{10}^{-7}}$ times the volume of the Sun. Express this figure in usual form.

Answer 1

Hint: First of all use the formula of the exponents given as ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$ to simplify the given expression. Now, to write the given scientific notation of the number in usual form, take the decimal point seven places towards the left of its position present in the number 7.67. Include six 0’s before the digit 7 that is present before the decimal point to get the answer.

Complete step-by-step solution:
Here we have been provided with the ratio of volume of earth to the volume of the sun in scientific notation as $7.67\times {{10}^{-7}}$. We are asked to write this expression in the usual form that is in the decimal form.
Now, let us assume the given ratio as R, so we have,
$\Rightarrow R=7.67\times {{10}^{-7}}$
Using the formula of exponents given as ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$ for any positive integer m we can rewrite the expression as: -
$\Rightarrow R=\dfrac{7.67}{{{10}^{7}}}$
We can see that the exponent of 10 in the denominator is 7, so to convert the above expression completely into the decimal form we have to move the decimal point seven places toward the left of its original place in the number present in the numerator, i.e. 7.67. We have to include six 0’s before 7 which is present before the decimal point. Therefore we get,
$\therefore R=0.000000767$
Hence, the volume of the earth is 0.000000767 times the volume of sun denoted in usual form.

Note: Note that the given ratio is in the standard form which is also called the scientific form or standard index form of representation of numbers. That numbers that are too big or too small are written in standard form to make calculations easy using the laws and certain formulas of exponents and powers. If the number would have been in the form $7.67\times {{10}^{7}}$ then we would have taken the decimal point seven places to the right of its position in 7.67.