\[{25^4} \div {5^3}\]. Simplify and express in exponential form.

Hint: Exponents are used to show repeated multiplication of a number by itself .For example $7 \times 7 \times 7 \times 7$ can be expressed as ${7^4}$ . Here the exponent is 4 which stands for the number of times the number 7 is multiplied and 7 is the base here which is the actual number that is getting multiplied. Here we will be using the concept and formulae of indices.

Complete step by step solution:
We can also write $25$ as $\left( {{5^2}} \right)$ then,
${25^4} \div {5^3} = {\left( {{5^2}} \right)^4} \div {5^3} - - - - (i)$
As, ${\left( {{a^m}} \right)^n} = {a^{mn}}$ then, from equation (i):
$
  {25^4} \div {5^3} = {5^{2 \times 4}} \div {5^3} \\
   = {5^8} \div {5^3} - - - - (ii) \\
 $
Also, ${a^m} \div {a^n} = {a^{m - n}}$ then, from equation (ii):
$
  {25^4} \div {5^3} = {5^{8 - 3}} \\
   = {5^5} \\
 $
Hence, \[{25^4} \div {5^3} = {5^5} = 3125\]


Note: In these types of problems we have to know the properties of the exponent functions and how to apply these laws in questions.
There are majorly six laws or rules defined for exponents and two mentioned in the solution.