Question

Find the value of \[{x^x}\]if \[{3^x}\;-{\text{ }}{3^{x - 1}}\; = {\text{ }}6\].
A. \[2\]
B. \[4\]
C. \[9\]
D. \[3\]

Answer 1

Hint: In this question, we need to find the value of \[{x^x}\]. For this, we have to simplify the given equation \[{3^x}\;-{\text{ }}{3^{x - 1}}\; = {\text{ }}6\] . After that, we will get the value of x by comparing the exponential coefficients. Based on the value of x, we can easily find the value of \[{x^x}\].

Formula used: The following exponential property is used to solve this question.
\[{a^{m - n}} = {a^m} \div {a^{ n}}\] and \[{{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}\]

Complete step-by-step solution:
We know that \[{3^x}\;-{\text{ }}{3^{x - 1}}\; = {\text{ }}6\]
Let us simplify the above equation.
But we know that \[{{a}^{m+n}}={{a}^{m}}\times {{a}^{n}}\]
By simplifying and by applying exponential property, we get
\[{3^x}\;-{\text{ }}{3^x} \times {3^{ - 1}}\; = {\text{ }}6\]
By taking \[{3^x}\;\]common from the above equation, we get
\[{3^x}\;\left( {1-{\text{ }}{3^{ - 1}}\;} \right) = 6\]
That means
\[{3^x}\;\left( {1-{\text{ }}\dfrac{1}{3}\;} \right) = 6\]
By simplifying, we get
\[{3^x}\;\left( {{\text{ }}\dfrac{{3 - 1}}{3}\;} \right) = 6\]
\[{3^x}\;\left( {{\text{ }}\dfrac{2}{3}\;} \right) = 6\]
\[{3^x}\; = \dfrac{{6 \times 3}}{2}\]
\[{3^x}\; = 9\]
But we can say that \[9 = 3 \times 3 = {3^2}\]
Thus, we get
\[{3^x}\; = {3^2}\]
Here, base is same. So, by comparing exponential coefficients (power), we get
\[x = 2\]
So, we get
\[{2^2} = 2 \times 2 = 4\]
Thus, the value of \[{x^x}\]is 4 if \[{3^x}\;-{\text{ }}{3^{x - 1}}\; = {\text{ }}6\].

Therefore, the correct option is (B).


Additional information: Once we merge numbers and variables in a viable way, using processes like addition, subtraction, multiplication, division, exponentiation, and other yet unlearned operations and functions, the resulting combination of mathematical symbols is known as a mathematical expression. Here, we can say that the exponents are used in exponential equations, as the name implies. We already understand that the exponent of a number (base) implies how many times the number (base) has been multiplied.

Note: Many students generally make mistakes in the simplification part of taking \[{3^x}\;\] common from the given equation such as \[{3^x}\;-{\text{ }}{3^{x - 1}}\; = {\text{ }}6\]. Thus, ultimately the end result may get wrong.