Question

Find the cube of $2.3\times {{10}^{7}}$.
(a) $1.2167\times {{10}^{22}}$
(b) $1.2167\times {{10}^{21}}$
(c) $1.2167\times {{10}^{26}}$
(d) $1.2167\times {{10}^{23}}$

Answer 1

Hint: In order to find the cube of any number of the type given in the question, firstly we need to assign to a variable, and find the cube of the number in the form of ${{a}^{3}}\times {{10}^{b}}$ where a is the number whose cube is to be find out and b is the power of 10 obtained. Now, separately, we need to find the cube of the number, i.e., ${{a}^{3}}$. Then, we simply need to update the value in the above obtained value. For doing these operations, we can use the properties of exponential functions or the properties of indexes.

Complete step-by-step answer:
Here, we have to find the cube of $\text{A}=2.3\times {{10}^{7}}\text{ (suppose)}.............(i)$. We will have to find the cube of the number in the form of ${{a}^{3}}\times {{10}^{b}}$ where a is the number whose cube is to be find out and b is the power of 10 obtained.
We have,
$\text{A}=2.3\times {{10}^{7}}$
Now, we take cube on both sides of the above equation, we get,
$\Rightarrow {{\left( \text{A} \right)}^{3}}={{\left( 2.3 \right)}^{3}}\times {{\left( {{10}^{7}} \right)}^{3}}$
Now, using the property of index, i.e., ${{\left( {{a}^{x}} \right)}^{y}}={{\left( a \right)}^{xy}}$, we get,
$\Rightarrow {{\left( \text{A} \right)}^{3}}={{\left( 2.3 \right)}^{3}}\times {{10}^{21}}...............(ii)$
Now, we separately need to find the cube of the number, i.e., ${{\left( 2.3 \right)}^{3}}$. So, we get,
${{\left( 2.3 \right)}^{3}}=2.3\times 2.3\times 2.3$
$\Rightarrow {{\left( 2.3 \right)}^{3}}=5.29\times 2.3$
$\Rightarrow {{\left( 2.3 \right)}^{3}}=12.167$
$\Rightarrow {{\left( 2.3 \right)}^{3}}=1.2167\times 10...............(iii)$
Then, we replace the value of the cube of ${{\left( 2.3 \right)}^{3}}$ from equation obtained at (iii) to equation (ii), we get,
$\Rightarrow {{\left( \text{A} \right)}^{3}}=1.2167\times 10\times {{10}^{21}}$
Now, using the property of index, ${{a}^{x}}\times {{a}^{y}}={{a}^{xy}}$, we get,
$\Rightarrow {{\left( \text{A} \right)}^{3}}=1.2167\times {{10}^{22}}$
Hence, the cube of $2.3\times {{10}^{7}}$ is $1.2167\times {{10}^{22}}$.

So, the correct answer is “Option A”.

Note: In these types of question where the cube is to be found out of the norm in the form $a\times {{10}^{b}}$, students normally make a mistake of multiplying the number with 10 and removing the term 10. This makes the question very difficult to solve. As here, we can see, if we multiply $2.3\times {{10}^{7}}$ we will get 23000000. Hence, we will have to find the cube of such a big number, which is very difficult and time consuming. Hence, students need to keep the term of power of 10 as it helps to solve the question. Besides, students need to be careful while moving the decimal place either ahead of a number or beyond a number. On moving a decimal beyond a number, the power of 10 decreases by 1 whereas on moving the decimal ahead of a number, the power of 10 increases by 1.