Question

Simplify $\dfrac{{{7^5} \cdot {7^3}}}{{{7^2}}}$?

Answer 1

Hint: This problem deals with solving the expressions with exponents and bases. An expression that represents repeated multiplication of the same factor is called a power. The number 7 is called the base, and the number of its power is called the exponent, on the left hand side of the given equation. The exponent corresponds to the number of times the base is used as a factor.
Here some basic rule of exponents and bases are used here such as:
$ \Rightarrow {a^m} \cdot {a^n} = {a^{m + n}}$
$ \Rightarrow \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$

Complete step by step answer:
The given expression is $\dfrac{{{7^5} \cdot {7^3}}}{{{7^2}}}$, which is considered below:
$ \Rightarrow \dfrac{{{7^5} \cdot {7^3}}}{{{7^2}}}$
Now consider the numerator of the given expression, as given below:
$ \Rightarrow {7^5} \cdot {7^3}$
Here applying the basic rule of ${a^m} \cdot {a^n} = {a^{m + n}}$, to the numerator, as shown below:
$ \Rightarrow {7^5} \cdot {7^3} = {7^{5 + 3}}$
$\therefore {7^5} \cdot {7^3} = {7^8}$
Now consider the denominator of the given expression, as given below:
$ \Rightarrow {7^2}$
There are no operations here in the denominator, as there is only one term.
Here applying the basic rule of $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$, to the numerator and the denominator, as shown below:
$ \Rightarrow \dfrac{{{7^8}}}{{{7^2}}} = {7^{8 - 2}}$
$ \Rightarrow \dfrac{{{7^8}}}{{{7^2}}} = {7^6}$
So the simplification of the given expression $\dfrac{{{7^5} \cdot {7^3}}}{{{7^2}}}$ is given below:
$\therefore \dfrac{{{7^5} \cdot {7^3}}}{{{7^2}}} = {7^6}$

Note: Please note that usually a power is represented with a base and an exponent. The base tells what number is being multiplied. The exponent, a small number written above and to the right of the base number, tells how many times the base number is being multiplied. The product rule says that to multiply two exponents with the same base, you keep the base and add the powers.
$ \Rightarrow {a^m} \cdot {a^m} = {a^{m + n}}$
$ \Rightarrow {\left( {{a^m}} \right)^n} = {a^{mn}}$
$ \Rightarrow {a^m} = {a^n}$, if the bases are the same, then the exponents have to be the same.