Question

Solve for value of $x$ , \[{3^{{x^2}}}:{3^x} = 9:1\]

Answer 1

Hint: Here we are asked to find the value of $x$ by using the given ratio \[{3^{{x^2}}}:{3^x} = 9:1\] . The ratio can be written in the fraction form then we will have an equation, then by simplifying that equation we can find the value of $x$ . During simplification we will be using some exponent formula to reduce the equation, those formulae are given in the formula sectio.
Formula:
Formula that we need to know:
$\dfrac{{{a^x}}}{{{a^y}}} = {a^{x - y}}$

Complete step by step answer:
It is given that \[{3^{{x^2}}}:{3^x} = 9:1\] from this we aim to find the value of $x$ .
Here it is given that two ratios are equal first, we will write the ratios in the form of fraction that is
$\dfrac{{{3^{{x^2}}}}}{{{3^x}}} = \dfrac{9}{1}$
Now let’s observe the left-hand side term it is in the form $\dfrac{{{a^x}}}{{{a^y}}}$ where the base $a = 3$ and the powers $x = {x^2}$ and $y = x$ . So, by using the formula $\dfrac{{{a^x}}}{{{a^y}}} = {a^{x - y}}$ we get
${3^{{x^2} - x}} = 9$
As we can see that the variable $x$ which we have to find is in the power to find its value we have to make the term on the other side as an exponent with base three.
The term of the right-hand side is nine, this can be factorized into $9 = 3 \times 3 = {3^2}$ now we have the modified the right-hand term as an exponent with base three. Thus, we have
${3^{{x^2} - x}} = {3^2}$
$ \Rightarrow {x^2} - x = 2$
$ \Rightarrow {x^2} - x - 2 = 0$
Let us solve this equation to find the value of $x$ using the formula $\dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Here $a = 1$, $b = - 1$ and $c = - 2$ substituting these in the formula we get
$x = \dfrac{{ - \left( 1 \right) \pm \sqrt {{{\left( 1 \right)}^2} - 4 \times 1 \times \left( { - 2} \right)} }}{{2 \times 1}}$
On simplifying the above, we get
$x = \dfrac{{ - 1 \pm \sqrt {1 + 8} }}{2}$
$x = \dfrac{{ - 1 \pm \sqrt 9 }}{2}$
$x = \dfrac{{ - 1 \pm 3}}{2}$
$x = \dfrac{{ - 1 + 3}}{2},x = \dfrac{{ - 1 - 3}}{2}$
$ \Rightarrow x = 2,x = - 1$
Thus, we have got the values of $x$ as $2$ and $ - 1$ .

Note:
A ratio is nothing but the comparison of two quantities of the same kind by a division. It is denoted by \[a:b\] and it can be written as \[\dfrac{a}{b}\] . In an equation when the bases are the same in the exponent terms on both sides then their power can be equated. That is why we have equated the powers \[{x^2} - x\] and the power two in the above calculation.