Question

The value of \[{\left( {625} \right)^{0.16}} \times {\left( {625} \right)^{0.09}}\] is
A) 4
B) 5
C) 25
D) 625

Answer 1

Hint:
Firstly, apply the property ${a^m} \times {a^n} = {a^{m + n}}$ on \[{\left( {625} \right)^{0.16}} \times {\left( {625} \right)^{0.09}}\] .
Thus, on solving the given condition using the above property step-by-step, we will get the required answer.

Complete step by step solution:
It is given that, \[{\left( {625} \right)^{0.16}} \times {\left( {625} \right)^{0.09}}\]
Now, applying the property ${a^m} \times {a^n} = {a^{m + n}}$ on \[{\left( {625} \right)^{0.16}} \times {\left( {625} \right)^{0.09}}\] .
 $\therefore {\left( {625} \right)^{0.16}} \times {\left( {625} \right)^{0.09}} = {\left( {625} \right)^{0.16 + 0.09}}$
$ = {\left( {625} \right)^{0.25}}$
Also, we can write 625 as ${5^4}$ and 0.25 as $\dfrac{1}{4}$.
So, ${\left( {625} \right)^{0.25}} = {\left( {{5^4}} \right)^{\dfrac{1}{4}}}$
Now, on applying the property ${\left( {{a^m}} \right)^n} = {a^{m \times n}}$ , we get
 $
{\left( {{5^4}} \right)^{\dfrac{1}{4}}} = {5^{4 \times \dfrac{1}{4}}} \\
= {5^1} \\
=5
$
Thus, \[{\left( {625} \right)^{0.16}} \times {\left( {625} \right)^{0.09}}\] =5.

So, option (B) is correct.

Note:
Some properties of exponents:
 $
  {a^m} \times {a^n} = {a^{m + n}} \\
  {a^n} \times {b^n} = {\left( {a \times b} \right)^n} \\
  \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} \\
  \dfrac{{{a^n}}}{{{b^n}}} = {\left( {\dfrac{a}{b}} \right)^n} \\
  {\left( {{a^m}} \right)^n} = {a^{m \times n}} \\
  \sqrt[n]{{{a^m}}} = {a^{\dfrac{m}{n}}} \\
  \sqrt[n]{a} = {a^{\dfrac{1}{n}}} \\
  {a^{ - n}} = \dfrac{1}{{{a^n}}} \\
  {a^0} = 1 \\
 $