Solve the Exponential Equation: 4^(x+8) = 1/256

We begin by addressing the problem as an equation:

$4^{?+8}=\frac{1}{256}$

Therefore, we replace the question mark with the unknown x and proceed to solve it:

$4^{x+8}=\frac{1}{256}$

Now let's briefly discuss the solution technique:

Generally speaking the goal when solving exponential equations is to reach a situation where there is a term on each side of the equation so that both sides have the same base. In such a situation we can unequivocally state that the power exponents on both sides of the equation are equal, and thus solve a simple equation for the unknown.

Mathematically, we will perform a mathematical manipulation (according to the laws of course) on both sides of the equation (or develop one side of the equation with the help of power rules and algebra) in order to reach the following situation as shown below:

$b^{m(x)}=b^{n(x)}$when$m(x),\hspace{4pt}n(x)$Algebraic expressions (actually functions of the unknown$x$) that can also exclude the unknowns ($x$) we are trying to find in the problem, which is the solution to the equation,

It is then stated that:

$m(x)=n(x)$and we can proceed to solve the simple equation that we obtained.

We return once again to solving the equation of the given problem:

$4^{x+8}=\frac{1}{256}$In solving this equation, various power rules are used:

a. Power rule with negative exponent:

$a^{-n}=\frac{1}{a^n}$b. Power rule for a power of an exponent raised to another exponent:

$(a^m)^n=a^{m\cdot n}$

Our goal initially is to simplify the terms of the equation, that is, "eliminate" fractions and roots (if there are any in the problem, there are none here)

In order to achieve this, we start by dealing with the fraction on the right side of the equation. This is done by using the power rule of a negative exponent specified in A above and represent this fraction (in parentheses) as a term with a negative exponent:

$4^{x+8}=\frac{1}{256} \\ 4^{x+8}=256^{-1}$Later on we want to be able to obtain the same base on both sides of the equation. The best way to achieve this is by breaking down each of the numbers in the problem into prime factors (using powers as well)

One can notice that both the number 256 and the number 4 are powers of the number 2, that is:

$4=2^2\\ 256=2^8$which is the presentation (decomposition) of the numbers 4 and 256 with the help of their prime factor, which is the number 2,

In this case although it is perhaps difficult to see directly that 256 is a power of 2 it can still be guessed because the only prime factor of the number 4 is the number 2 and therefore it makes sense that 256 is also a power of 2.

The above value can be also be verified by inserting it into different powers and checking if indeed the number 256 is obtained,

Thus we again return to the equation that we obtained in the last step and proceed replace these numbers in their initial factorization:

We use parentheses due to the fact that the number being replaced is a number in power,

In the next step, we apply the power rule for an exponent set in B above in order to eliminate the parentheses. We will do this for the terms on both sides of the equation step by step for each side. Each step is seen below:

$\big(2^2\big)^{x+8}= \big(2^8\big)^{-1} \\ 2^{2\cdot (x+8)}= \big(2^8\big)^{-1} \\ 2^{2\cdot (x+8)}= 2^{8\cdot (-1)} \\ 2^{2\cdot (x+8)}= 2^{-8} \\$We develop each section separately step by step. In the last step we simplify the expression in the exponent on the right side of the equation. On the left side the exponent can be further simplified using the distributive property despite there being no obligation to do so at this stage since we can leave it for later.

We have reached our goal, we have obtained an equation in which both sides have terms with the same base, therefore we can state that the exponents of the terms on both sides are equal, and to solve the resulting equation for the unknown we will do the following:

$2^{2\cdot (x+8)}= 2^{-8} \\ \\ \downarrow\\ 2 (x+8)=-8$We continue and solve the resulting equation, we will do this by opening the parentheses with the help of the distributive property and isolating the unknown on the left side, we will achieve this in the usual way: moving the sections and dividing the final equation by the coefficient of the unknown:

$2 (x+8)=-8 \\ 2x+16=-8\\ 2x=-8-16\\ 2x=-24 \hspace{8pt}\text{/:}(2) \\ \frac{\not{2}x}{\not{2}}=\frac{-24}{2}\\ x=-\frac{24}{2}\\ \bm{x=-12 }$In the first step we open the parentheses on the left side using the distributive property, then we simplify the equation by moving the sections, remembering that when a term switches sides it changes sign, and finally we complete the isolation of the unknown by dividing both sides of the equation by its factor and simplify the resulting expression by reducing the fractions,

We have thus solved the given equation. Below is a brief summary of the solution steps:

$4^{x+8}=\frac{1}{256} \\ 4^{x+8}=256^{-1}\\ \big(2^2\big)^{x+8}= \big(2^8\big)^{-1} \\ 2^{2\cdot (x+8)}= 2^{-8} \\ \\ \downarrow\\ 2 (x+8)=-8 \\ 2x+16=-8\\ 2x=-24 \hspace{8pt}\text{/:}(2) \\ \bm{x=-12 }$Therefore, the correct answer is option c.