Chapter 7.3, Problem 101PFA

To determine

To find the solution of the equation.

Answer to Problem 101PFA

  $x=1$

Explanation of Solution

Given:

Equation: ${16}^{x+2}=2^{12}$

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,

  ${16}^{x+2}=2^{12}$

Here, 16 can be written as $2^4.$

Hence,

  $\begin{array}{c}$ {(2^4)}^{x+2}=2^{12}$$ 2^{4(}{}^{x+2)}=2^{12}[\because {(a^b)}^c=a^{bc}]$4(x+2)=12[\because a^{f(x)}=a^{g(x)},\text{then }f(x)=g(x)]\end{array}$

Simplifying the above linear equation,

  $4(x+2)=12$

Applying distributive property,

  $\begin{array}{c}\left(4\times x\right)+\left(4\times 2\right)=12\\ 4x+8=12\end{array}$

Subtracting 8on both sides and simplifying,

  $\begin{array}{c}4x+8-8=12-8\\ 4x=4\end{array}$

Dividing 4 on both sides,

  $\begin{array}{c}\frac{4x}{4}=\frac{4}{4}\\ x=1\end{array}$

Conclusion:

Therefore, the solution to the given equation is $x=1.$