Question

How do you solve $\dfrac{{{{81}^{3x + 2}}}}{{{{243}^x}}} = 3$?

Answer 1

Hint: First prime factorizes the base of each term. After that use the law of exponent, ${\left( {{{\left( a \right)}^b}} \right)^c} = {a^{bc}}$. Then apply the law, $\dfrac{{{a^b}}}{{{a^c}}} = {a^{b - c}}$ on the left side of the expression. After that equate the exponent as the base of both sides will be the same. Then do simplification to get the desired result.

Complete step-by-step solution:
Exponents and powers are the two mathematical terms used to simplify the problems, especially in algebra. Hypothetically, both powers and exponents are synonymous, but in mathematical relationships, they are used in different contexts. Power is a core mathematical expression that is used to represent exactly how many times a number should be utilized in a given multiplication. On the other hand, exponents are either positive or negative numbers which represent the power to which the base number is raised. To make it simple to understand, power is an expression that represents repeated multiplication of the same number whereas exponent refers to a quantity that represents the power to which the number is raised.
We can write ${81^{3x + 2}}$ as,
$ \Rightarrow {81^{3x + 2}} = {\left( {{3^4}} \right)^{3x + 2}}$
We know that,
${\left( {{{\left( a \right)}^b}} \right)^c} = {a^{bc}}$
Using the rules, we get
$ \Rightarrow {81^{3x + 2}} = {3^{4\left( {3x + 2} \right)}}$
Multiply the terms in the exponent,
$ \Rightarrow {81^{3x + 2}} = {3^{12x + 8}}$
We can also write ${243^x}$ as,
$ \Rightarrow {243^x} = {\left( {{3^5}} \right)^x}$
Using the above rules, we get
$ \Rightarrow {243^x} = {3^{5x}}$
So, the expression can be written as,
$ \Rightarrow \dfrac{{{3^{12x + 8}}}}{{{3^{5x}}}} = 3$
We know that,
$\dfrac{{{a^b}}}{{{a^c}}} = {a^{b - c}}$
Using the above rules on the left side, we get
$ \Rightarrow {3^{12x + 8 - 5x}} = 3$
Simplify the terms,
$ \Rightarrow {3^{7x + 8}} = 3$
As the base of both sides of the expression is the same. So, equate the exponent of the expression,
$ \Rightarrow 7x + 8 = 1$
Move the constant term on the right side,
$ \Rightarrow 7x = 1 - 8$
Subtract the value on the right side,
$ \Rightarrow 7x = - 7$
Divide both sides by 7,
$\therefore x = - 1$

Hence, the value of $x$ is -1.

Note: The conceptual knowledge about exponents and laws of exponents is required. Students should always keep in mind various laws of exponents to solve these types of questions. Mistakes can be by students while applying the law of exponent.