How do you simplify: $\dfrac{{{5^4} \times {5^7}}}{{{5^8}}}$ ?

Hint: We will first use the formula which states that: ${x^a} \times {x^b} = {x^{a + b}}$ and find the numerator of given expression. Then we will just use the formula: $\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$.

Complete step-by-step solution:
We are given that we are required to simplify $\dfrac{{{5^4} \times {5^7}}}{{{5^8}}}$.
Let us assume that $A = \dfrac{{{5^4} \times {5^7}}}{{{5^8}}}$ ………….(1)
We know that we have a formula given by the following expression:-
$ \Rightarrow {x^a} \times {x^b} = {x^{a + b}}$
Replacing x by 5, a by 4 and b by 7 and putting all of these in the above mentioned formula, we will then obtain the following expression with us:-
$ \Rightarrow {5^4} \times {5^7} = {5^{4 + 7}}$
Simplifying the calculation on the right hand side of the above expression, we will then obtain the following expression:-
$ \Rightarrow {5^4} \times {5^7} = {5^{11}}$
Putting this above mentioned result in the equation number 1, we will then obtain the following equation:-
$ \Rightarrow A = \dfrac{{{5^{11}}}}{{{5^8}}}$ ………………(2)
Now, we also know that we have a formula given by the following expression:-
$ \Rightarrow \dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$
Replacing x by 5, a by 11 and b by 8 and putting all of these in the above mentioned formula, we will then obtain the following expression with us:-
$ \Rightarrow \dfrac{{{5^{11}}}}{{{5^8}}} = {5^{11 - 8}}$
Simplifying the calculation on the right hand side of the above expression, we will then obtain the following expression:-
$ \Rightarrow \dfrac{{{5^{11}}}}{{{5^8}}} = {5^3}$
Putting this above mentioned result in the equation number 2, we will then obtain the following equation:-
$ \Rightarrow A = {5^3}$

Hence, the required answer is $\dfrac{{{5^4} \times {5^7}}}{{{5^8}}} = {5^3}$.

Note: The students must notice that they may also further simplify the expression by multiplying 5 three times which is equal to 125.
Hence, the required answer will be $\dfrac{{{5^4} \times {5^7}}}{{{5^8}}} = 125$.
The students must note that if you do not want to use the formulas which we have used in the solution above, then we also have an alternate way given by:-
We are given that we are required to simplify $\dfrac{{{5^4} \times {5^7}}}{{{5^8}}}$.
We can write this expression as:-
\[ \Rightarrow \dfrac{{{5^4} \times {5^7}}}{{{5^8}}} = \dfrac{{\left( {5 \times 5 \times 5 \times 5} \right) \times \left( {5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5} \right)}}{{\left( {5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5} \right)}}\]
Now, we can remove the brackets to write this as:-
\[ \Rightarrow \dfrac{{{5^4} \times {5^7}}}{{{5^8}}} = \dfrac{{5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5}}{{5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5 \times 5}}\]
Now, we can write this further as:-
\[ \Rightarrow \dfrac{{{5^4} \times {5^7}}}{{{5^8}}} = 5 \times 5 \times 5 = 125\]