Chapter 7.3, Problem 80PPS

To determine

To solve the given equation.

Answer to Problem 80PPS

  $x=3$

Explanation of Solution

Given:

Equation: ${81}^{2x-3}=9^{x+3}$

Formula used:

Exponent rules:

  ${(a^b)}^c=a^{bc}$

  $a^{f(x)}=a^{g(x)},$ then $f(x)=g(x)$

Distributive property: $a(b+c)=a\cdot b+a\cdot c$

Calculation:

Consider the given equation,

  $\Rightarrow {81}^{2x-3}=9^{x+3}$

Here, 81 can be written as $9^2.$

Hence,

  $\begin{array}{l}$ {(9^2)}^{2x-3}=9^{x+3}$$ 9^{2(2}{}^{x-3)}=9^{x+3}[\because {(a^b)}^c=a^{bc}]$2(2x-3)=(x+3)[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $2(2x-3)=(x+3)$

Applying distributive property to the left-hand side,

  $\begin{array}{c}(2\times 2x)-(2\times 3)=(x+3)\\ 4x-6=x+3\end{array}$

Subtracting x on both sides and simplifying,

  $\begin{array}{c}4x-6-x=x+3-x\\ 3x-6=3\end{array}$

Adding 6 on both sides and simplifying,

  $\begin{array}{c}3x-6+6=3+6\\ 3x=9\end{array}$

Dividing 3 on both sides and simplifying,

  $\begin{array}{c}\frac{3x}{3}=\frac{9}{3}\\ x=3\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=3.$