Question

Value of ${\left( {256} \right)^{0.16}} \times {\left( {256} \right)^{0.09}}$ is
$\left( a \right){\text{ 4}}$
$\left( b \right){\text{ 16}}$
$\left( c \right){\text{ 64}}$
$\left( d \right){\text{ 256}}{\text{.25}}$

Answer 1

Hint:
As we can see in the question that the base is the same and the power is different. So by using one of the properties of power we will solve it. The property is ${x^a} \times {x^b} = {x^{a + b}}$ . So by using this we will solve the question equation and we will get the solution.

Formula used:
Multiplicative property of the power given by
${x^a} \times {x^b} = {x^{a + b}}$
Here,
$x$ , will be the base
And $a,b$ will be the power of the base.

Complete Step by Step Solution:
So we have the question given as ${\left( {256} \right)^{0.16}} \times {\left( {256} \right)^{0.09}}$ .
Since the base of the question is the same and the power is different so by using them, we will get
$ \Rightarrow {\left( {256} \right)^{0.16}} \times {\left( {256} \right)^{0.09}}$
Now on comparing with the formula we can write the above equation as
$ \Rightarrow {\left( {256} \right)^{0.16}} \times {\left( {256} \right)^{0.09}} = {\left( {256} \right)^{0.16 + 0.09}}$
And on solving the addition in the power section, we get
$ \Rightarrow {\left( {256} \right)^{0.25}}$
In terms of the fraction, the power can be written as
$ \Rightarrow {\left( {256} \right)^{\dfrac{{25}}{{100}}}}$
Now on solving the power, as they are divisible so we get the equation as
$ \Rightarrow {\left( {256} \right)^{\dfrac{1}{4}}}$
As we know that the number $256$ can be written as ${4^4}$ .
Therefore, on substituting this, we will get
$ \Rightarrow {\left( {{4^4}} \right)^{\dfrac{1}{4}}}$
So the power will get multiplied here,
$ \Rightarrow {4^1}$
And on solving it, we will get
$ \Rightarrow 4$

Hence, the option $\left( a \right)$ is correct.

Note:
For solving this type of question all we need to do is make the equation or we can say power or either base in such a way that it follows one of the properties of power and then we can easily solve it by using some mathematical operations. So the use of properties will strike in mind when we will do practice on questions related to it. So by practice, we can solve this type of question easily.