Question

Solve \[{{9}^{x+2}}=720+{{9}^{x}}\].

Answer 1

Hint: We have to solve the equation for the value of x. So for that, first, we will make the base of all the numbers equal in the equation and then we will compare their powers. It should also be noted that if we are multiplying two numbers with the same base then their powers will be added.

Complete step by step answer:
We have to solve the equation \[{{9}^{x+2}}=720+{{9}^{x}}\] and we have to find the value of x in this equation.
In the above equation, each number is having some power but \[720\] is not having any power on it, so first, we will convert it in the form of power. So that solving the question will be easy for us. We will now do the factors of \[720\] which are as shown below.
\[720=2\times 2\times 2\times 2\times 3\times 3\times 5\]
As the equation contains the base nine so we will also make \[720\] in the form of base nine.
\[\begin{align}
  & 720={{2}^{4}}\times {{3}^{2}}\times 5 \\
 & \Rightarrow 720={{3}^{2}}\times 80 \\
 & \Rightarrow 720={{3}^{2}}\times 10\times 8 \\
\end{align}\]
As we know, if we add one to eight then the result will be nine and if we subtract one from ten then the result will be nine. So writing ten and eight in the form of that only.
\[720={{3}^{2}}\times (9+1)(9-1)\]
As nine is the square of three. So we will write nine as the power of three.
\[720={{3}^{2}}({{3}^{2}}+1)({{3}^{2}}-1)\]……eq(1)
As we know that, \[(a-b)(a+b)={{a}^{2}}-{{b}^{2}}\]
So applying this identity in eq(1), we get the following results.
\[720={{3}^{2}}({{3}^{4}}-1)\]…..eq(2)
We have the equation,
\[{{9}^{x+2}}=720+{{9}^{x}}\]
On taking the terms with base nine on one side and the constant term on the other side. We get.
\[{{9}^{x+2}}-{{9}^{x}}=720\]
On putting the value of \[720\] from eq(2), we get the following results.
\[{{9}^{x+2}}-{{9}^{x}}={{3}^{2}}({{3}^{4}}-1)\]
On writing this equation in the form of power of three, the following results will be obtained which are as follows.
\[\begin{align}
  & {{(3)}^{2(x+2)}}-{{3}^{2x}}={{3}^{2}}({{3}^{4}}-1) \\
 & \Rightarrow {{3}^{2x+4}}-{{3}^{2x}}={{3}^{2}}({{3}^{4}}-1) \\
\end{align}\]
On taking \[{{3}^{2x}}\] common on the left-hand side, we get the following results.
\[{{3}^{2x}}({{3}^{4}}-1)={{3}^{2}}({{3}^{4}}-1)\]
On canceling the like terms.
\[{{3}^{2x}}={{3}^{2}}\]
As the bases are equal, so their powers will also be equal. Hence, after comparing the powers we get.
\[\begin{align}
  & 2x=2 \\
 & \Rightarrow x=1 \\
\end{align}\]
So the value of x is \[1\] and hence the equation is solved.

Note:
Every number contains two parts. The base part and the exponent part. The base part represents the actual number and the exponent part tells us how many times that number will be multiplied. If a number has power equals zero, then the result will always be equal to one.