Question

If \[{2^x} \times {4^x} = {8^{\dfrac{1}{3}}} \times {\left( {32} \right)^{\dfrac{1}{5}}}\], find the value of \[x\].

Answer 1

Hint:
Here we will firstly convert the base of the equation in terms of the exponents of 2. Then we will simplify the equation. Then we will compare the powers of the equation and equate them. We will then solve the equation to get the value of \[x\].

Complete step by step solution:
Given equation is \[{2^x} \times {4^x} = {8^{\dfrac{1}{3}}} \times {\left( {32} \right)^{\dfrac{1}{5}}}\].
Firstly we will write the numbers in the form of the exponents of 2.
We know that 4 can be written as \[{2^2}\], 8 can be written as \[{2^3}\] and 32 can be written as \[{2^5}\]. So we will write this in the main equation, we get
\[ \Rightarrow {2^x} \times {\left( {{2^2}} \right)^x} = {\left( {{2^3}} \right)^{\dfrac{1}{3}}} \times {\left( {{2^5}} \right)^{\dfrac{1}{5}}}\]
Now simplifying this equation, we get
\[ \Rightarrow {2^x} \times {2^{2x}} = {2^{\dfrac{{3 \times 1}}{3}}} \times {2^{\dfrac{{5 \times 1}}{5}}}\]
\[ \Rightarrow {2^{x + 2x}} = {2^1} \times {2^1}\]
We know this algebraic property that when the base is the same then the exponents get added i.e. \[{x^a}{x^b} = {x^{a + b}}\]. Therefore, we get
\[ \Rightarrow {2^{3x}} = {2^2}\]
Now by comparing the powers of the equation, we get
\[ \Rightarrow 3x = 2\]
Dividing both sides by 3, we get
\[ \Rightarrow x = \dfrac{2}{3}\]

Hence, the value of \[x\] is \[\dfrac{2}{3}\].

Note:
Here, we have the exponents of the number in the fraction form so to make it in the normal form we have to convert the number in the exponents of 2. Exponent is defined as the number which represents how many times a number, is being multiplied to itself. If the exponent of a number is zero then the value of the number is 1 i.e. \[{a^0} = 1\]. We need to also keep in mind that the multiplication of a number to its reciprocal is equal to 1.