Question

If $x = \left( {123456789} \right)\left( {76543211} \right) + {\left( {23456789} \right)^2}$, then find the number of zeros in $\sqrt[4]{x}$?
A. $3$
B. $4$
C. $5$
D. $1$

Answer 1

Hint: In the above question you have given the value of $x$ and find the number of zeros in $\sqrt[4]{x}$. In order to solve this, you have to use basic maths and some tricks. First you have to split the first term in the value of $x$ and apply the algebraic identity which is given by $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$ and then further solve it, you’ll get the answer.

Formula used:
The algebraic identity used in this question is given by
$\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$

Complete step-by-step answer:
Here, in the question we have given the value of $x$ as
$x = \left( {123456789} \right)\left( {76543211} \right) + {\left( {23456789} \right)^2}$
Now, we can rewrite it as
$x = \left( {100000000 + 23456789} \right)\left( {100000000 - 23456789} \right) + {\left( {23456789} \right)^2}$ ……..(i)
Now here, we can apply the algebraic identity which is given by
$\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$
On applying the identity in the equation (i), the value of $x$ becomes
$ \Rightarrow x = {(100000000)^2} - {(23456789)^2} + {(23456789)^2}$
Here the positive-negative terms cancel out with each other so, we get
$ \Rightarrow x = {\left( {100000000} \right)^2}$
On further solving it, we get
$ \Rightarrow x = {({10^8})^2} = {10^{16}}$
On taking power of $\dfrac{1}{4}$ on the both sides, we get
$ \Rightarrow {x^{\dfrac{1}{4}}} = {({10^{16}})^{\dfrac{1}{4}}}$
On further solving, we get the value of $\sqrt[4]{x}$ as,
$ \Rightarrow {x^{\dfrac{1}{4}}} = {10^{16 \times \dfrac{1}{4}}}$
On further solving the above relation, we get
$\sqrt[4]{x} = {10^4}$
Hence, the number of zeros in $\sqrt[4]{x}$ is $4$.

Therefore, the correct option for this question is (B).

Note:
These algebraic equations that are valid for all the values of the variables in the equations are known as algebraic identities. Basically, identity means that the left-hand side of the equation is identically equal to the right-hand side, for all values of the variables.
These algebraic identities are also used for the factorization of polynomials. In this way, algebraic identities are used in the computation of algebraic expressions and solving different polynomials. Also, note that all the standard algebraic identities are basically derived from the Binomial theorem.