Question

Given
${5}^{x} - {5}^{x - 2} = 3000.$
So find the value of
${(x)}^{x}$
​

Answer 1

let 5^x be y

y-y/25=3000

24y=75000

y=3125

5^x=3125 (y=5^x)

5^x=5^5

x=5

Now,

x^x=5^5==3125

Answer 2

$\huge{\underline{\green{\rm{Solution:}}}}$

$\small{\underline{\red{\rm{Given:}}}}$

• $\rm{\:{5}^{x}-{5}^{x-2}=3000}$

$\small{\underline{\purple{\rm{To\:Find:}}}}$

• $\rm{The\: value of (x)^x}$

$\small{\underline{\red{\rm{According\:to\:Given\: expression:}}}}$

$\rm{\implies {5}^{x}-{5}^{x}*{5}^{-2}=3000}$

• $\rm{by\:taking\:common\:{5}^{x}\: here,}$

$\rm{\implies {5}^{x}(1-{5}^{-2})=3000}$

$\rm{\implies {5}^{x}(1-\dfrac{1}{25})=3000}$

$\rm{\implies {5}^{x}(\dfrac{25-1}{25})=3000}$

$\rm{\implies {5}^{x}(\dfrac{24}{25})=3000}$

$\rm{\implies {5}^{x}*\cancel{24}=\cancel{3000}*25}$

$\rm{\implies {5}^{x}=125*25}$

$\rm{\implies {5}^{x}={5}^{3}*{5}^{2}}$

$\rm{\implies {5}^{x}={5}^{3+2}}$

$\rm{\implies {\cancel{5}}^{x}={\cancel{5}}^{5}}$

$\rm\purple{\implies x=5}$

Hence,

$\rm{\implies The\: value\:of\:({x})^{x}}$

$\rm{\implies The\: value\:of\:({x})^{x}=5^5}$

$\rm\purple{\implies The\: value\:of\:({x})^{x}=3125}$

$\small{\underline{\red{\rm{Some\: formula\: related\:to\: this\:Question:}}}}$

$\rm{\implies {a}^{m}*{a}^{n}={a}^{m+n}}$

$\rm{\implies {a}^{-m}=\dfrac{1}{a^m}}$

$\rm{\implies ({a}^{m})^{n}={a}^{mn}}$