Question

If an algebraic expression is given as \[{{9}^{x+2}}=240+{{9}^{x}}\], then x =
(A). 0.5
(B). 0.2
(C). 0.4
(D). 0.1

Answer 1

Hint: Write the equation with both the left-hand side and the right-hand side. Now assume \[{{9}^{x}}\] as a new variable after simplifying the left-hand side by using properties of powers. Now you get an equation with variables on both sides of the equation. Try to bring all variables on one side. Subtract the variable term on the right-hand side on both sides of the equation. Now you have a variable only on the left-hand side. Find the coefficient of the variable on the left hand side. Divide the equation on both sides with this co – efficient. By this you get the value of the variable. Substitute \[{{9}^{x}}\] back and then find the value of x which is the required result.

Complete step-by-step solution -
Given equation in terms of x, can be written as follows:
\[\Rightarrow {{9}^{x+2}}=240+{{9}^{x}}\]
By properties of power, we know the formulas as:
\[\Rightarrow {{a}^{b+c}}={{a}^{b}}.{{a}^{c}}\]
By using the above formula on left hand side, we get:
\[\Rightarrow {{9}^{x}}{{.9}^{2}}=240+{{9}^{x}}\]
By simplifying the above equation, we can write as:
\[\Rightarrow {{81.9}^{x}}=240+{{9}^{x}}\]
So, we have the term \[{{9}^{x}}\] on both sides of equations.
We need the value of x. So, for our convenience we assume that the term \[{{9}^{x}}\] is a variable denoted by t.
By substituting t into the equation above, we get it as:
\[\Rightarrow 81t=240+t\]
The co – efficient of t on the right hand side id written by 1.
 So, by subtracting the term t on both sides, we get it as:
\[\Rightarrow 81t-t=240+t-t\]
By simplifying and cancelling the common terms, we get it as:
\[\Rightarrow 80t=240\]
As the value of constant on the Right-hand side is 240, we move to the next step, coefficient of variable on the left-hand side is 80.
By dividing with 80 on both sides of equation, we get:
\[\Rightarrow \dfrac{80t}{80}=\dfrac{240}{80}\]
By simplifying this equation, we get the value of t as:
\[\Rightarrow t=3\]
By substituting the value of t in expression $t= {9}^{x}$, we get it as:
\[\Rightarrow {{9}^{x}}=3\]
As we know \[{{3}^{2}}=9\] we can write this equation: \[{{3}^{2x}}={{3}^{1}}\].
So, we get it as: 2x = 1. So, \[x=\dfrac{1}{2}=0.5\].
Therefore option (a) is correct.

Note: Be careful while using properties of power as the whole solution depends on them. We can solve without assuming any new variable directly by using \[{{9}^{x}}\]. But for clarity here we assumed a new variable instead of using \[{{3}^{2}}\] we can say \[x={{\log }_{9}}3\]. By logarithmic properties, we can find x as 0.5.