Question

Solve for x

Please explain

I will mark you as brainlist

Please answer it very fast

Answer 1

$\underline{\large{\mathfrak{Solution : }}}$


$\mathsf{ \implies {3}^{ \normalsize{(4x \: + \: 1)}} \: = \: \left( \dfrac{1}{9} \right)^{ \normalsize{x \: + \: 1} }} \\ \\ \mathsf{ \implies {3}^{ \normalsize{(4x \: + \: 1)}} \: = \: \left( \dfrac{1}{ {3}^{2} } \right)^{ \normalsize{x \: + \: 1}} }$



$\underline{\underline{\mathsf{Using \: algebraic \: identity : }}} \\ \\ \boxed{\mathsf{\implies \dfrac{1}{a^{m}} \: = \: {a}^{ \normalsize{( - m)}}}}$


$\mathsf{ \implies {3}^{ \normalsize{(4x \: + \: 1)}} \: = \: \left[3^{ \normalsize{(-2)}} \right]^{\normalsize{(x \: + \: 1)} }}$



$\underline{\underline{\mathsf{Using \: algebraic \: identity : }}} \\ \\ \boxed{\mathsf{\implies (a^{ \normalsize{m}})^{ \normalsize{n}} \: = \: a^{ \normalsize{mn}}}}$



$\mathsf{\implies 3^{\normalsize{(4x \: + \: 1)}} \: = \: 3^{\normalsize{(-2x \: - \: 2)}}}$



$\underline{\underline{\textsf{On \: Comparison : }}} \\ \\ \mathsf{\implies 4x \: + \: 1 \: = \: - 2x \: - \: 2 } \\ \\ \mathsf{\implies 4x \: + \: 2x \: = \: -2 \: - \: 1} \\ \\ \mathsf{\implies 6x \: = \: -3 } \\ \\ \mathsf{\implies x \: = \: \dfrac{-3}{6}} \\ \\ \mathsf{\implies x \: = \: \dfrac{-1}{2}}$



$\boxed{\underline{\underline{\large{\mathsf{ The \: value \: of \: x \: is \: \dfrac{-1}{2}.}}}}}$

Answer 2

Hey there!

Given equation to us $\bf{3^{4x + 1} = (\frac{1}{9})^{x + 1}}$

Converting the fractional value of $\bf{(\frac{1}{9})^{x + 1}}$ to the base of value "3" into denominator by power to negative value of "2" that is,

$\bf{(\frac{1}{9})^{x + 1} = (\frac{1}{3^2})^{x + 1}) = (3^{- 2})^{x + 1}}$

$\bf{3^{4x + 1} = (3^{- 2})^{x + 1}}$

Applying the rule of exponential powers that is:

$\bf{(a^b)^c = a^{b \times c}}$.

Here,

$\bf{(3^{- 2})^{x + 1} = 3^{- 2(x + 1)}}$

Therefore,

$\bf{3^{4x + 1} = 3^{- 2(x + 1)}}$

Now, apply the function powering rule to cancel out similar bases to obtain the equation values that is,

$\bf{a^{f(x)} = a^{g(x)}, \: \: then \qquad f(x) = g(x)}$

Therefore,

$\bf{4x + 1 = - 2(x + 1)}$

Now, solve this equation for finding exact value for variable "x":

Expand this equation of  - 2(x + 1) by applying the principles of distributive law that is,  

a(b + c) = ab + ac,    Here,       a = -2,  b = x  and  c = 1.

And apply the basic "plus-minus rules" that is,    +(- a) = - a  and Multiply the numbers:

$\bf{4x + 1 = - 2x - 2}$

Subtracting the value of "1" from both the provided sides to us, we get; By Simplification:

$\bf{4x + 1 - 1 = - 2x - 2 - 1}$

$\bf{4x = - 2x - 3}$

Adding the variable joined value of "2x" to both the sides:

$\bf{4x + 2x = - 2x - 3 + 2x}$

Simplify the equation;

$\bf{6x = - 3}$

Divide both the following sides with the value of "6" and simplify the equation:

$\bf{\dfrac{6x}{6} = \dfrac{- 3}{6}}$

$\bf{x = \dfrac{- 3}{6}}$

Apply the rule of fractional shifting of negative sign and cancel out the common factor of "3";

Therefore the final value and the required answer becomes:

$\boxed{\bf{x = - \dfrac{1}{2}}}$

Which is the required solution for this type of query.

Hope this helps you and solves your doubts for this query to find the variable values!