Question

9th Grade solution please
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Answer 1

Answer:

D is a right answer ddddd

Answer 2

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

$\rm :\longmapsto\: {5}^{x - 1} = {25}^{x + 1}$

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

The value of x

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

Given that

$\rm :\longmapsto\: {5}^{x - 1} = {25}^{x + 1}$

We know, that

$\boxed{ \rm{ {x}^{m + n} = {x}^{m} \times {x}^{n}}}$

and

$\boxed{ \rm{ {x}^{m - n} = {x}^{m} \div {x}^{n}}}$

So, using this identity, the given expression reduced to

$\red{\rm :\longmapsto\: {5}^{x} \div {5}^{1} = {25}^{x} \times {25}^{1}}$

$\red{\rm :\longmapsto\: {5}^{x} \div {5}^{1} = {(5 \times 5)}^{x} \times {5 \times 5}^{}}$

$\rm :\longmapsto\: \dfrac{ {5}^{x} }{5} = {5}^{2x} \times {5}^{2}$

$\rm :\longmapsto\: {5}^{x} = {5}^{2x} \times {5}^{2} \times 5$

We know that,

$\boxed{ \rm{ {x}^{m + n} = {x}^{m} \times {x}^{n}}}$

So, using this we get

$\rm :\longmapsto\: {5}^{x} = {5}^{2x + 2 + 1}$

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

We know,

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

So, we get

$\rm :\longmapsto\:x = 2x + 3$

$\rm :\longmapsto\:x - 2x = 3$

$\rm :\longmapsto\:- x = 3$

$\bf\implies \:x = - \: 3$

• So, Option A) is correct.

Verification :-

Given expression is

$\rm :\longmapsto\: {5}^{x - 1} = {25}^{x + 1}$

On substituting x = - 3, we get

$\rm :\longmapsto\: {5}^{ - 3 - 1} = {25}^{ - 3 + 1}$

$\rm :\longmapsto\: {5}^{ - 4} = {25}^{ - 2}$

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

$\rm :\longmapsto\: {5}^{ - 4} = {5}^{ - 2 \times 2}$

$\rm :\longmapsto\: {5}^{ - 4} = {5}^{ -4}$

Hence, Verified

Alternative Method :-

Given expression is

$\rm :\longmapsto\: {5}^{x - 1} = {25}^{x + 1}$

can be rewritten as

$\rm :\longmapsto\: {5}^{x - 1} = {(5 \times 5)}^{x + 1}$

$\rm :\longmapsto\: {5}^{x - 1} = {5}^{2(x + 1)}$

$\rm :\longmapsto\: {5}^{x - 1} = {5}^{2x + 2}$

We know,

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

So, using this, we get

$\rm :\longmapsto\:x - 1 = 2x + 2$

$\rm :\longmapsto\:x - 2x = 1 + 2$

$\rm :\longmapsto\:- x = 3$

$\rm :\longmapsto\:x = - \: 3$