Find the values of $x$ in each of the following:$ 2^{x-7} \times 5^{x-4}=1250 $

Given:

\( 2^{x-7} \times 5^{x-4}=1250 \)

To do: 

We have to find the value of $x$.

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

$2^{x-7} \times 5^{x-4}=1250$

$\Rightarrow 2^{x} \times 2^{-7}\times5^{x}\times5^{-4}=2 \times 5 \times 5 \times 5 \times 5$

$\Rightarrow \frac{2^{x} \times 5^{x}}{2^{7} \times 5^{4}}=2 \times 5 \times 5 \times 5 \times 5$

$\Rightarrow(10)^{x}=2 \times 5 \times 5 \times 5 \times 5 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 5 \times 5 \times 5 \times 5$

$\Rightarrow (10)^{x}=2^{8} \times 5^{8}$

$\Rightarrow (10)^{x}=(10)^{8}$

Comparing both sides, we get,

$x=8$

The value of $x$ is $8$.    

Tutorialspoint

Updated on: 10-Oct-2022

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