Question

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If (125) ^ (2x + 4) = (3125) ^ (2x - 4) , then the value of x is:​

Answer 1

$\large\underline{\sf{Given- }}$

$\rm :\longmapsto\: {125}^{2x + 4} = {3125}^{2x - 4}$

$\blue{\large\underline{\sf{To\:Find - }}}$

The value of x

$\red{\large\underline{\sf{Solution-}}}$

Given expression is

$\rm :\longmapsto\: {125}^{2x + 4} = {3125}^{2x - 4}$

$\rm :\longmapsto\: {(5 \times 5 \times 5)}^{2x + 4} = {(5 \times 5 \times 5 \times 5 \times 5)}^{2x - 4}$

$\rm :\longmapsto\: {5}^{3(2x + 4)} = {5}^{5(2x - 4)}$

$\rm :\longmapsto\: {5}^{6x + 12} = {5}^{10x - 20}$

We know that

$\red{\rm :\longmapsto\: {x}^{m} = {x}^{n} \: \rm \implies\:m = n \: }$

So, using this

$\rm :\longmapsto\:6x + 12 = 10x - 20$

$\rm :\longmapsto\:6x - 10x = - 12 - 20$

$\rm :\longmapsto\: - 4x = -32$

$\rm \implies\:x = 8$

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Learn More :-

$\begin{gathered}(1)\:{\underline{\boxed{\bf{\blue{a^m\times{a^n}\:=\:a^{m\:+\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(2)\:{\underline{\boxed{\bf{\purple{\dfrac{a^m}{a^n}\:=\:a^{m\:-\:n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(3)\:{\underline{\boxed{\bf{\orange{\dfrac{1}{x^n}\:=\:x^{-n}\:}}}}} \\ \end{gathered}$

$\begin{gathered}(4)\:{\underline{\boxed{\bf{\red{ {x}^{0} = 1}}}}} \\ \end{gathered}$

Answer 2

Given:

(125)⁽²ˣ⁺⁴⁾=(3125)⁽²ˣ⁻⁴⁾

To Find : Value of x

Solution:

(125)⁽²ˣ⁺⁴⁾=(3125)⁽²ˣ⁻⁴⁾

125 = 5 * 5 * 5 = 5³

3125 = 5 * 5 * 5 *  5 * 5 = 5⁵

=> (5³)⁽²ˣ⁺⁴⁾=( 5⁵)⁽²ˣ⁻⁴⁾

Using law of exponent (xᵃ)ᵇ = xᵃᵇ

=> 5⁽⁶ˣ⁺¹²⁾=5⁽¹⁰ˣ⁻²⁰⁾

Equate the powers as bases are same

=> 6x + 12 = 10x - 20

=> 4x = 32

=> x = 8

Hence the value of x = 8

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