Question

If we have an expression \[{27^x} = \dfrac{9}{{{3^x}}}\] find \[x\]

Answer 1

Hint: Here we have to find the value of \[x\].For this, first we write \[{27^x}\] and \[9\] as a power of \[3\] so that the whole equation is converted to a single base form. After converting to a single base, we use the basic rules of exponent and some laws for further simplification.
The rules used are:
(1) If the bases are equal, we can equate the powers.
\[ \Rightarrow {a^m} = {a^n}\]
\[ \Rightarrow m = n\]
The laws used are:
(1) Power of a power:
\[ \Rightarrow {\left( {{a^m}} \right)^n} = {a^{m \times n}} = {a^{mn}}\]
(2) Quotient with same bases:
\[ \Rightarrow \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]
where a is a non-zero integer and m and n are integers.

Complete step-by-step solution:
It is given that, \[{27^x} = \dfrac{9}{{{3^x}}}{\text{ }} - - - \left( 1 \right)\]
And we have to find the value of \[x\]
Since, \[27\] can be written as \[{3^3}\]
\[ \Rightarrow {27^x} = {\left( {{3^3}} \right)^x}\]
And \[9\] can be written as \[{3^2}\]
Thus, equation \[\left( 1 \right)\] can be written as,
\[{\left( {{3^3}} \right)^x} = \dfrac{{{3^2}}}{{{3^x}}}{\text{ }} - - - \left( 2 \right)\]
By using power of a power law on the left side of the equation \[\left( 2 \right)\] ,
i.e., \[{\left( {{a^m}} \right)^n} = {a^{m \times n}} = {a^{mn}}\]
here, \[a = 3,{\text{ }}m = 3,{\text{ }}n = x\]
\[ \Rightarrow {\left( {{3^3}} \right)^x} = {3^{3 \times x}} = {3^{3x}}\]
\[\therefore \] equation \[\left( 2 \right)\] can be written as,
\[\left( {{3^{3x}}} \right) = \dfrac{{{3^2}}}{{{3^x}}}{\text{ }} - - - \left( 3 \right)\]
Now by using dividing powers with the same base law on the right side of the equation \[\left( 3 \right)\]
i.e., \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]
here, \[a = 3,{\text{ }}m = 2,{\text{ }}n = x\]
\[ \Rightarrow \dfrac{{{3^2}}}{{{3^x}}} = {3^{2 - x}}\]
\[\therefore \] equation \[\left( 3 \right)\] can be written as,
\[{3^{3x}} = {3^{2 - x}}{\text{ }} - - - \left( 4 \right)\]
Bases are equal, so we can equate the powers
\[ \Rightarrow 3x = 2 - x\]
\[ \Rightarrow 3x + x = 2\]
On simplification, we get
\[4x = 2\]
\[ \Rightarrow x = \dfrac{2}{4}\]
On cancelling, we get
\[x = \dfrac{1}{2}\]
Hence, we get the value of \[x = \dfrac{1}{2}\].

Note: When we are faced with an equation with exponents, think about each law of exponent. Start by identifying which rules we need to use to solve the equation. Then, we can quickly determine the best way to simplify the problem and can find a solution. These laws are applicable for negative powers also.
Some useful laws are:
(1) Product of Power
\[ \Rightarrow {a^m} \times {a^n} = {a^{m + n}}\]
(2) Power of a Product
\[ \Rightarrow {\left( {ab} \right)^m} = {a^m} \times {b^m}\]
(3) Power of a Quotient
 \[ \Rightarrow {\left( {\dfrac{a}{b}} \right)^m} = \dfrac{{{a^m}}}{{{b^m}}}\]
(4) Zero Power
\[ \Rightarrow \left( {{a^0}} \right) = 1\]
(5) Negative Power
\[ \Rightarrow {a^{ - m}} = \dfrac{1}{{{a^m}}}\]
where a is a non-zero integer and m and n are integers.