Question

If ${{2}^{x-7}}\times {{5}^{x-4}}=1250$ then find the value of x.

Answer 1

Hint: First of all to solve this equation we have calculated the prime factorization of 1250 and arrange it in power of the multiple of 2 and 5. After that arrange and compare the powers of 2 and 5 separately on both sides and solve the equations obtained and get the value of x.

Complete step-by-step answer:
 Given that
${{2}^{x-7}}\times {{5}^{x-4}}=1250$ ….. (1)
Now we have to calculate the prime factorization of 1250 i.e.
$1250=2\times 5\times 5\times 5\times 5$
Also written in terms of powers
i.e. $1250={{2}^{1}}\times {{5}^{4}}$
Putting this value in the equation (1)
We get ${{2}^{x-7}}\times {{5}^{x-4}}={{2}^{1}}\times {{5}^{4}}$
Comparing the powers of 2 and 5, we get
$\begin{align}
  & {{2}^{x-7}}={{2}^{1}} \\
 &\Rightarrow x-7=1 \\
 & \Rightarrow x=8 \\
\end{align}$
$\begin{align}
  & {{5}^{x-4}}={{5}^{4}} \\
 &\Rightarrow x-4=4 \\
 & \therefore x=8 \\
\end{align}$
Hence, we are getting equal value of x i.e. $x=8$.

Note: As in equation (2) the RHS of equation represents the prime factorization of the number on the LHS of the equation.
The prime factors of a number are all the prime numbers that, when multiplied together, equal the original number. We can find the prime factorization of a number by using a factor tree and dividing the number into smaller parts.