Question

If‌ ‌we have an expression as \[{2^x}‌ ‌=‌ ‌{3^y}‌ ‌=‌ ‌{12^z}\],‌ ‌show‌ ‌that‌ ‌\[\dfrac{1}{z}‌ ‌=‌ ‌\dfrac{1}{y}‌ ‌+‌ ‌\dfrac{2}{x}\].‌

Answer 1

Hint: We have to prove the given expression using the given expression \[{2^x} = {3^y} = {12^z}\]. We solve this question using the concept of the common ratios of the terms and the concept of the exponents rules and properties. We will first equate the given expression to a constant variable and then write the value of each given term in terms of the considered constant, then we will simply the expression which we have to prove such that on substituting the values of the considered constant back into the expression we get the required answer.

Complete step-by-step solution:
Given:
\[{2^x} = {3^y} = {12^z}\]
To prove:
\[\dfrac{1}{z} = \dfrac{1}{y} + \dfrac{2}{x}\]
Proof:
Let us consider that \[k\] is a constant term such that, we can write the expression as:
\[{2^x} = {3^y} = {12^z} = k\]
Now splitting the equation for each term, we can write the expression as:
\[{2^x} = k\] , \[{3^y} = k\] and \[{12^z} = k\]
Also, simplifying each expression we get
\[2 = {k^{\dfrac{1}{x}}}\] , \[3 = {k^{\dfrac{1}{y}}}\] and \[12 = {k^{\dfrac{1}{z}}}\]
Now we will represent the terms as a product of each other as:
\[12 = 2 \times 2 \times 3\]
\[12 = {2^2} \times 3\]
Now substituting the value of the constant back into the above expression, we can wite the expression as:
\[{k^{\dfrac{1}{z}}} = {\left( {{k^{\dfrac{1}{x}}}} \right)^2} \times {k^{\dfrac{1}{y}}}\]
We know that the exponents rule of product is as given below:
\[{({b^m})^n} = {b^{m \times n}}\]
Using the rule of exponents, we get the expression as:
\[{k^{\dfrac{1}{z}}} = {k^{\dfrac{2}{x}}} \times {k^{\dfrac{1}{y}}}\]
Also, we know that the exponents rule for product of terms with same bases is given as:
\[{a^m} \times {a^n} = {a^{m + n}}\]
Using the rule of exponents, we get the expression as:
\[{k^{\dfrac{1}{z}}} = {k^{\dfrac{2}{x} + \dfrac{1}{y}}}\]
We also know that the exponents rule of bases states that when the bases of two variables are same then their powers are equal.
i.e. if \[{a^n} = {a^m}\]
then \[n = m\]
As both the left hand side and the right hand side of the expression have the same base element I.e. \[k\].
Hence, Using the rule of exponents, we get the expression for the value of the expression as:
\[\dfrac{1}{z} = \dfrac{2}{x} + \dfrac{1}{y}\]
Hence proved

Note: The base of a number is known as the number which is raised to a power.
The various rules and properties for the exponents are as given below:
Product rule: \[{a^n} \times {b^n} = {(a \times b)^n}\]
Quotient rule: \[\dfrac{{{a^n}}}{{{a^m}}} = {a^{n - m}}\]
Zero rule: \[{b^0} = 1\]
One rule: \[{b^1} = b\]
This is the easiest way to do solve such problems by using the properties of exponents as without properties we can’t simplify the expression.