Question

Say true or false and justify your answer: $10\times {{10}^{11}}={{100}^{11}}$

Answer 1

Hint: Solve the left-hand side of the given expression and the right-hand side of it as well. Now, give justification by relating their values. Use the property of surds with the problem wherever required.
Complete step-by-step answer:
Two properties can be given as
$\begin{align}
  & {{m}^{a}}\times {{m}^{b}}={{m}^{a+b}} \\
 & {{\left( {{m}^{a}} \right)}^{b}}={{m}^{a\times b}} \\
\end{align}$

As the expression given in the problem is
$10\times {{10}^{11}}={{100}^{11}}........(i)$
Now, to justify the statement in equation (i) we need to simplify the left-hand side and right-hand side of the equation and hence if after simplifying them, both the simplified values are same, then equation is justified, otherwise not.
So, let us just solve the Left-hand side of the expression i.e. \[10\times {{10}^{11}}\].So, LHS can be given as
$LHS=10\times {{10}^{11}}......(ii)$
Now, we can observe that the power of the first 10 involved in the multiplication is 1 and the power of another 10 is 11. So, we can use the identity of surds as
${{m}^{a}}\times {{m}^{b}}={{m}^{a+b}}......(iii)$
Now, compare the LHS of the above equation with the equation(ii) and hence apply the relation (iii). So, we can observe that the value of m is 10, a = 1, b = 11. It means, we can write the equation (ii) as
$\begin{align}
  & LHS={{10}^{1}}\times {{10}^{11}}={{10}^{1+11}}={{10}^{12}} \\
 & \Rightarrow LHS={{10}^{12}}........(iv) \\
\end{align}$
Now, let us simplify the relation given in Right hand side of equation (i). So, we have RHS of equation as
$RHS={{\left( 100 \right)}^{11}}........(v)$
As we know 100 can be replaced with the term 10 x 10 or ${{10}^{2}}$ in the above expression. So, we get
$RHS={{\left( {{\left( 10 \right)}^{2}} \right)}^{11}}........(vi)$
Now, we can use property of surds with the above equation given as
${{\left( {{m}^{a}} \right)}^{b}}={{m}^{a\times b}}........(vii)$
Hence, we can compare the LHS of equation (vii) and the equation (vi). So, we get m = 10, a = 2, b = 11. Hence, using the identity of equation (vii) with equation (vi), we get
$\begin{align}
  & RHS={{\left( {{\left( 10 \right)}^{2}} \right)}^{11}}={{\left( 10 \right)}^{2\times 11}}={{10}^{22}} \\
 & \Rightarrow RHS={{10}^{22}}........(viii) \\
\end{align}$
Now, we can observe the equation (iv) and (viii) and get that value of LHS and RHS of the equation (i) is not equal to each other as ${{10}^{12}}\ne {{10}^{22}}$ .Hence the statement given in the problem is false.

Note: Another approach for solving the given problem would be that we can find exact values of LHS by multiplying 10 to 11 times and RHS by multiplying 100 to 22 times. But it is not very flexible as given in the solution. We will get the same answer with it as well.
One may get confused with the property of surds for the expression ${{m}^{a}}\times {{m}^{b}}={{m}^{a+b}}$ .One may apply it as ${{m}^{a}}\times {{m}^{b}}={{m}^{a-b}}$ by confusing with the identity $\dfrac{{{m}^{a}}}{{{m}^{b}}}={{m}^{a-b}}$ .So, be clear with the identities of surds to make these problems easier and very flexible
One may multiply the terms of LHS as $10\times {{10}^{11}}={{100}^{11}}$ , which is wrong and hence, he/she will get incorrect answers. So, we cannot multiply the numbers if powers of them are different. We need to simplify them for getting multiplication of them. So, don’t confuse it with this.