Question

solve :
$49 ^{x + 4 } \\ = 7 ^{2} (343)^{x + 1}$
​

Answer 1

Answer :-

Value of x is 3.

Explanation :-

$\mathsf{ 49^{x + 4} = 7^2(343^{x + 1}) }$

$\mathsf{ \implies ( 7^2) ^{x + 4} = 7^2 \{( 7^3) ^{x + 1} \} }$

$\mathsf{ \implies 7^{2(x + 4)} = 7^2 \{( 7^{3(x + 1)} \} }$

$\boxed{ \bf \because ( {a}^{m})^n = {a}^{mn} }$

$\mathsf{ \implies 7^{2x + 8} = 7^2 ( 7^{3x + 3} ) }$

$\mathsf{ \implies 7^{2x + 8} = 7^{2 + 3x + 3} }$

$\boxed{ \bf \because {a}^{m} \times a^n = {a}^{m + n} }$

$\mathsf{ \implies 7^{2x + 8} = 7^{3x + 5} }$

$\mathsf{ \implies 2x + 8 = 3x + 5}$

$\boxed{ \bf if \ \because {a}^{m} = a^n \ then \ m = n}$

$\mathsf{ \implies 8 - 5= 3x - 2x}$

$\mathsf{ \implies 3 = x}$

$\mathsf{ \implies x = 3}$

∴ the value of x is 3.

Answer 2

Given

${49}^{(x + 4)} = {7}^{2} \times {343}^{(x + 1)}$

To find

The value of x

EXPLANATION

$= > {49}^{(x + 4)} = {7}^{2} \times {343}^{(x + 1)}$

$= > {7}^{2 ^ {(x + 4)} } = {7}^{2} \times {7}^{3 ^{(x + 1)}}$

$\underline {\bf {a^{m}}^{n}=a^{mn}}$

Using the formula

$= > {7}^{2(x + 4)} = {7}^{2} \times {7}^{3(x + 1)}$

$= > {7}^{(2x + 8)} = {7}^{2} \times {7}^{(3x + 3)}$

$\underline{\bf a^m×a^n=a^{m+n}}$

$= > {7}^{(2x + 8)} = {7}^{(2 + 3x + 3)} \\ \\ = > {7}^{(2x + 8)} = {7}^{(3x + 5)}$

$\underline{\bf a^m=a^n \:then \: m=n}$

Equating the powers

$= > 2x + 8 = 3x + 5$

$= > - x = - 3$

$= > x = 3$

$\boxed {\sf x=3}$

Important formulas

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

$= > \dfrac{ {a}^{m} }{ {a}^{n} } = {a}^{m - n}$

$= > {a}^{0} = 1$

$=>a^{-m}=\dfrac{1}{a^m}$