Question

Express in simplest exponential form:
\[25\times 125\].

Answer 1

Hint: There are various rules that are applied on numbers and their powers to find the values in an easier way.
The numbers and their powers are connected by four arithmetic operations – addition, subtraction, multiplication and division.

Complete step by step answer:
The rules on base and power of numbers are as follows:
a^m × aⁿ = (a)^{m + n}
a^m ÷ aⁿ = (a)^{m - n}
(ab)^m = a^m × b^m
a^m × b^m = (ab)^m
${{a}^{0}}=1$
The base “a” and “b” can be a whole number or a rational number and the same applies to power also.
Similarly, bases and powers can be negative or positive. This indicates that both bases and powers belong to rational numbers as rational numbers include all types of integers, zero and both positive and negative fractions.
The rules related to base and powers help in calculating complex problems in very less time.
Simple exponential form is given by having the base as a non-squared number.
Expression: \[25\times 125\]can be written in simple form as:
\[25\times 125={{(5)}^{2}}\times {{(5)}^{3}}\]
The number twenty-five is square of five while one hundred twenty five is the cube of five.
The rule related to same base and different power is applied to get simple exponential form as follows:
 \[25\times 125={{(5)}^{2}}\times {{(5)}^{3}}={{(5)}^{2+3}}={{(5)}^{5}}\]

Note: The power of a number denotes the number of times that number has to be multiplied with itself.
For instance, ${{\left( 2 \right)}^{3}}$ denotes that number $2$ must be multiplied $3$ times to get answer equal to $2\times 2\times 2$ equal to $8$ and similarly if $8$ is to be written as power of some number, it is written as ${{(2)}^{3}}$.