Question

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

Answer 1

Hint: According to given in the question we have to determine the solution of the expression or we can say that we have to solve the expression $\dfrac{{{5^4} \times {5^7}}}{{{5^8}}}$. So, first of all we have to multiply the terms of the expression which can be multiplied with the help of the formula which is as mentioned below:

Formula used: $ \Rightarrow {a^m} \times {a^n} = {a^{m + n}}...............(A)$
According to the formula (A) of the numbers which are same or we can say that if the bases are same then on multiplying such number the given powers if that number will be added to each other.
Now, we have to divide the terms of the expression which can be divided with the help of the formula which is as explained below:
$ \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}...............(B)$
According to the formula (B) of the numbers which are same or we can say that if the bases are same then on multiplying such number the given powers if that number will be subtracted to each other.
Now, we just have to solve the power of the number obtained after solving the given expression.

Complete step by step answer:
Step 1: First of all we have to multiply the terms of the expression which can be multiplied with the help of the formula (A) which is as mentioned in the solution hint. Hence,
$
   \Rightarrow \dfrac{{{5^{4 + 7}}}}{{{5^8}}} \\
   \Rightarrow \dfrac{{{5^{11}}}}{{{5^8}}} \\
 $
Step 2: Now, we have to divide the terms of the expression which can be divided with the help of the formula (B) which is as explained in the solution hint. Hence,
$
   \Rightarrow {5^{11 - 8}} \\
   \Rightarrow {5^3}
 $
Step 3: Now, we just have to solve the power which is the cube of the number obtained in the solution step 2 after solving the given expression. Hence,
 $
   \Rightarrow {5^3} = 5 \times 5 \times 5 \\
   \Rightarrow {5^3} = 125
 $

Hence, with the help of the formula (A), and (B) we have simplified the expression $\dfrac{{{5^4} \times {5^7}}}{{{5^8}}}$ whose value is $125$.

Additional information:
The GCF of two numbers is equal to the common prime factors of both the numbers. The prime factors of the number are the numbers that can divide the number without leaving a remainder.

Note: If the given numbers are different to each other then we can’t add or subtract the power of that number and if the number are same or we can say that if the bases are same then on multiplying those number power will be added and on dividing those numbers powers will be subtracted.
To obtain the value of the expression we can also find the powers of all the given numbers and after then we have to just multiply and divide the terms.