Question

If we have an expression as \[{(\sqrt 3 )^5} \times {9^2} = {3^n} \times 3\sqrt 3 \], then what is the value of $n$?

Answer 1

Hint: First, we need to know about the concept of the square root. The square root of the number is a value, which on multiplied by itself given the original number, which is the given numbers that obtain by multiplying any of the whole numbers (zero to infinity) twice, or the square of the given numbers yields a whole number like \[\sqrt 9 = {\sqrt {3^2}} = 3\] and also $\sqrt 3 = {3^{\dfrac{1}{2}}}$

Complete step-by-step solution:
Since from the problem given that \[{(\sqrt 3 )^5} \times {9^2} = {3^n} \times 3\sqrt 3 \] and we need to find the unknown value of the $n$
Let us convert the given equation into some simplified form so that it is easy to find the unknown value.
Since we know that $\sqrt 3 = {3^{\dfrac{1}{2}}}$ substitute this form in the given equation we have \[{({3^{\dfrac{1}{2}}})^5} \times {9^2} = {3^n} \times 3({3^{\dfrac{1}{2}}})\]
Using the concept of the power rule we have ${a^m}{a^n} = {a^{m + n}}$ and ${({a^m})^n} = {a^{mn}}$
Thus, apply these values in the given we get ${({3^{\dfrac{1}{2}}})^5} \times {9^2} = {3^n} \times 3({3^{\dfrac{1}{2}}})$
$ \Rightarrow {3^{\dfrac{5}{2}}} \times {9^2} = {3^n} \times 3({3^{\dfrac{1}{2}}})$
Since ${3^2} = 9$ then we have,
${3^{\dfrac{5}{2}}} \times {9^2} = {3^n} \times 3({3^{\dfrac{1}{2}}})$
$\Rightarrow {3^{\dfrac{5}{2}}} \times {3^4} = {3^n} \times 3({3^{\dfrac{1}{2}}})$
Further solving we get,
${3^{\dfrac{5}{2} + 4}} = {3^n} \times {3^{\dfrac{1}{2} + 1}}$
$\Rightarrow {3^{\dfrac{{13}}{2}}} = {3^{n + \dfrac{3}{2}}}$ and thus the base values are same hence we have ${a^m} = {a^n} \Rightarrow m = n$.
Therefore, we get,
 ${3^{\dfrac{{13}}{2}}} = {3^{n + \dfrac{3}{2}}}$
$ \Rightarrow \dfrac{{13}}{2} = n + \dfrac{3}{2}$
Now put the variable on the left side and the numbers on the right side we get
\[n = \dfrac{{13}}{2} - \dfrac{3}{2} = \dfrac{{10}}{2} = 5\]
Hence the value of the unknown is $n = 5$.

Note: $\sqrt {} $ is known as the radical symbol. We used the concept of power terms, square and square root, and also power rules like ${a^m}{a^n} = {a^{m + n}}$ and ${({a^m})^n} = {a^{mn}}$
We are also able to check whether the answer is correct or not.
Let us substitute the value of the $n = 5$ in the given equation then we have \[{(\sqrt 3 )^5} \times {9^2} = {3^5} \times 3\sqrt 3 \]
Since both the equations are the same, let us simplify further to prove that \[{3^{\dfrac{5}{2}}} \times {9^2} = {3^5} \times 3({3^{\dfrac{1}{2}}})\]
Again, make use of the power rules formula we get
\[{3^{\dfrac{5}{2} + 4}} = {3^5} \times {3^{\dfrac{1}{2} + 1}} \]
\[ \Rightarrow {3^{\dfrac{{13}}{2}}} = {3^{5 + \dfrac{3}{2}}}\]
Finally, we get the answer as $\dfrac{{13}}{2} = \dfrac{{13}}{2}$ and hence both are same values and also $n = 5$ is the correct answer.