Question

maths
$125 ^x = \frac{25}{5 {}^{x} } \: \: how \: many \: value \: of \: x$
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Answer 1

★ Given :

$125 ^x = \dfrac{25}{5^x}$

★ To find :

Value of x

★ Solution :

25 = 5²

125 = 5 × 5 × 5 = 5³

by putting the exponential values :

$\implies (5^3)^x = \dfrac{ 5^2}{5^x}$

★ Law of exponents -

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

$\leadsto \frac{a^m}{a^n} = a^{m - n}$

by applying the laws -:

$\implies 5^{3x} = 5 ^{2 - x}$

→ If the base is equal , then the powers also become equal.

3x = 2 - x

3x + x = 2

4x = 2

$\purple{\leadsto x = \frac{1}{2}}$

★ Additional Information :

Laws of exponents -

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

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

$\implies \frac{a^m}{a^n} = a^{m - n}$

$\implies a^0 = 1$

$\implies \frac{1}{a^m} = a^{-m}$