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: In the given expression, apply the property ${a^m} \times {a^n} = {a^{m + n}}$ and then convert 625 in term of powers of 5. Once, this is done, all you need to do is to apply the power rule of exponents to get the required answer.

Complete step-by-step answer:
Let us begin by considering the given expression, which is;
${\left( {625} \right)^{0.16}} \times {\left( {625} \right)^{0.09}}$
As the exponents are involved with the same bases and different powers, we can simplify the given expression by using the property;
${a^m} \times {a^n} = {a^{m + n}}$
In the expression given to us, we have;
$
  a = 625 \\
  m = 0.16 \\
  n = 0.09 \\
 $
Using the exponents property we get;
$
  {\left( {625} \right)^{0.16}} \times {\left( {625} \right)^{0.09}} = {\left( {625} \right)^{0.16 + 0.09}} \\
   = {\left( {625} \right)^{0.25}} \\
 $
Now, we need to simplify the obtained expression further. For that, we use the fact that;
${5^4} = 625$.
Thus,
$ \Rightarrow {\left( {625} \right)^{0.25}} = {\left( {{5^4}} \right)^{0.25}}$
It is known, by the power rule of exponents that, ${\left( {{a^m}} \right)^n} = {a^{m \cdot n}}$ , thus we get;
$
  {\left( {{5^4}} \right)^{0.25}} = {5^{4 \cdot \left( {0.25} \right)}} \\
   = {5^{1.00}} \\
   = {5^1} \\
   = 5 \\
 $
Hence, the given expression’s value is equal to 5. Thus, option (B) is the correct option.

Note: Always use the properties of exponents to first simplify the expressions involving powers. If that is not done, and you went with the option of finding the values using calculator or any other tool, you might not as well get the correct accurate answer. Always look at the options given to you in the question, mostly the hint is actually there in the options, like in this one it was understood that the answer would be some power of 5, so we tried converting 625 in powers of 5 in order to get rid of the decimal exponents.