Question

The number ${\left( {1024} \right)^{1024}}$ is obtained by raising ${\left( {{{16}^{16}}} \right)^n}$ . What is the value of n ?
$\left. {\text{A}} \right)64$
$\left. {\text{B}} \right){64^2}$
$\left. {\text{C}} \right){64^{64}}$
$\left. {\text{D}} \right)160$

Answer 1

Hint:We must have basic knowledge about exponents and it’s properties to solve this question. In the term ${x^2}$, $x$ is called the base and $2$ is known as the exponent ( also called power ) . The exponent part represents the number of times the base part has to be multiplied, hence exponent is an alternative way of representing repeated multiplication. For example: ${x^2} = x \times x$ , means $x$ is multiplied twice.

Complete step-by-step solution:
The given question is ;
$ \Rightarrow {\left( {1024} \right)^{1024}} = {\left[ {{{\left( {16} \right)}^{16}}} \right]^n}$
We have to find the value of n , so using the exponents rules and properties we have to simplify the expression in such a way that we easily get the value of n .
Simplifying the given expression, we get;
$ \Rightarrow {\left( {{2^{10}}} \right)^{1024}} = {\left( {16} \right)^{16n}}$
Using the exponents of exponents property i.e. ${\left( {{x^m}} \right)^p} = {x^{mp}}$ ;
It can be further simplified as ;
$ \Rightarrow {\left( 2 \right)^{10240}} = {\left( {{2^4}} \right)^{16n}}$
$ \Rightarrow {\left( 2 \right)^{10240}} = {\left( 2 \right)^{64n}}$
As both the L.H.S. and R.H.S. have same base values , hence their respective powers can be compared;
$ \Rightarrow 64n = 10240$
From the above equation, we get the value of n as ;
$ \Rightarrow n = \dfrac{{10240}}{{64}}$
$\therefore n = 160$
Therefore, the correct answer for this question is option D i.e. $160$ .

Note: Some of the important exponents formulae are listed here which can be used to solve the similar type of question: $\left( 1 \right)$ ${x^0} = 1$ , value for zero exponent. $\left( 2 \right){x^1} = x$ , value for unit exponent . $\left( 3 \right)\sqrt x = {x^{\dfrac{1}{2}}}$ . $\left( 4 \right)\sqrt[n]{x} = {x^{\dfrac{1}{n}}}$ , value for a fractional exponent . $\left( 5 \right){x^{ - n}} = \dfrac{1}{{{x^n}}}$ , value for negative exponent. $\left( 6 \right){x^n} = \dfrac{1}{{{x^{ - n}}}}$ . $\left( 7 \right){x^m}{x^n} = {x^{m + n}}$ , adding the exponents . $\left( 8 \right)\dfrac{{{x^m}}}{{{x^n}}} = {x^{m - n}}$ , subtracting the exponents . $\left( 9 \right){\left( {{x^m}} \right)^p} = {x^{mp}}$ , exponents of exponents .