Question

find the value of x if
$625^{x} = \frac{25}{5 ^{x} }$
​

Answer 1

Hey there!

First, let's simplify both the sides.

625 = 5⁴ ; 625ˣ = 5⁴ˣ

25 can be written as 5²; 5²/5ˣ = 5²⁻ˣ

a^n / a^m = a^(n-m)

We can now write this equation as:

5⁴ˣ = 5²⁻ˣ

Cancelling the bases, we get ( 4x = 2 - x )

Now, we will have to solve this.

5x = 2

x = 2 / 5

Answer 2

$\Large \mathfrak {\underline {\underline{ Solution : }}} \\ \\ \sf 625^{x} = \frac{25}{5 ^{x} } \: \: \: \: \\ \sf {({5}^{4})}^{x} = \frac{ {5}^{2} }{ {5}^{x} } \: \: \: \: \: \: \: \: \: \: \: \: \{ {({x}^{a})} ^{b} = x {}^{ab} \} \\ \sf {5}^{4x} = {5}^{ 2 - x} \: \: \: \: \: \: \: \: \: \: \: \: \: \bigg \{ \frac{ {a}^{x} }{ {a}^{y} } = {a}^{x - y} \bigg \} \\ \\ \textsf{With Same Base, the Power be equal} \\ \implies \sf 4x = 2 - x \\ \implies \sf 4x + x = 2 \\ \implies \sf x = \frac{2}{5}$