Question

Solve for $X$:
(A) ${{6}^{X}}=\dfrac{1}{216}$
(B) ${{2}^{X}}={{4}^{2X+1}}$

Answer 1

Hint: Here we have been given two equations and we have to find the value of an unknown variable in each case. Firstly as the unknown variable is in power of the exponent in each case so we will make the base same of both sides by simplifying the terms. Then we will compare the exponent on both sides and form an equation. Finally we will solve the equation and get the desired answer.

Complete step by step answer:
We have been given two equations as follows:
(A) ${{6}^{x}}=\dfrac{1}{216}$
(B) ${{2}^{X}}={{4}^{2X+1}}$
Solving them one by one we get,
(A) ${{6}^{X}}=\dfrac{1}{216}$
Now as we can see that the base of the exponent on left side is $6$ so we will write the term on the right side as an exponent with base $6$ as follows,
As $216=6\times 6\times 6$
$\Rightarrow 216={{6}^{3}}$
Put the above value in the equation we get,
${{6}^{X}}=\dfrac{1}{{{6}^{3}}}$
Now as we know the negative exponent law which is ${{a}^{-m}}=\dfrac{1}{{{a}^{m}}}$ using it above we get,
$\Rightarrow {{6}^{X}}={{6}^{-3}}$
Comparing the power both side as base is same, so we get,
$\Rightarrow X=-3$
Hence $X=-3$ is the solution for equation ${{6}^{X}}=\dfrac{1}{216}$ .

(B) ${{2}^{X}}={{4}^{2X+1}}$
As we can see that the base of the exponent on the left side is $2$ and on the right side is $4$ we will simplify the right side as follows,
As $4={{2}^{2}}$
Put the above value in the equation,
$\Rightarrow {{2}^{X}}={{\left( {{2}^{2}} \right)}^{2X+1}}$
$\Rightarrow {{2}^{X}}={{2}^{2\left( 2X+1 \right)}}$
As we know by the rule of exponent that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$ using it above we get,
$\Rightarrow {{2}^{X}}={{2}^{4X+2}}$
Comparing the power both side as base is same, so we get,
$\Rightarrow X=4X+2$
$\Rightarrow X-4X=2$
On solving,
$\Rightarrow -3X=2$
$\Rightarrow X=-\dfrac{2}{3}$
Hence $X=-\dfrac{2}{3}$ is the solution for equation ${{2}^{X}}={{4}^{2X+1}}$ .

Note:
Exponents are used to represent large numbers in a simplified manner. Power is used to express the multiplication of the same number multiple times. In this type of questions the most important step is to make the base of the whole equation the same so that we can compare them and get an equation without an exponent which on solving gives us our solution.