The $\displaystyle x$-intercept of a function is where $\displaystyle y = 0$. Thus, we are looking for the $\displaystyle x$-value which makes $\displaystyle 0 = 4^{x}$.

If we try to solve this equation for $\displaystyle x$ we get an error.

To bring the exponent down we will need to take the natural log of both sides.

$\displaystyle ln(0)=xln(4)$

Since the natural log of zero does not exist, there is no exponent which makes this equation true.

Thus, there is no $\displaystyle x$-intercept for this function. 

Exponential functions with a base greater than one are models of exponential growth. Thus, we know that our function will increase and not decrease. Remembering the graph of an exponential function, we can determine that the graph will begin gradually, almost like a flat line. Then, as $\displaystyle x$ increases, $\displaystyle {}y$ begins to increase very quickly. 

The base of the right side can be rewritten as base two.

$\displaystyle 2^{2x+3} = 2^{3(-6x)}$

$\displaystyle 2x+3 = 3(-6x)$

Simplify the equation.

$\displaystyle 2x+3 = -18x$

Add $\displaystyle 18x$ on both sides.

$\displaystyle 2x+3 +18x= -18x+18x$

$\displaystyle 20x+3 = 0$

Subtract 3 on both sides.

$\displaystyle 20x+3 -3= 0-3$

$\displaystyle 20x = -3$

Divide by 20 on both sides.

 $\displaystyle \frac{20x}{20} = \frac{-3}{20}$

The answer is:  $\displaystyle -\frac{3}{20}$

To solve, change the base of the fraction to base 4.

$\displaystyle \frac{1}{4} = 4^{-1}$

$\displaystyle 4^{2x-1} = 4^{-1}$

With similar bases, we can set the powers equal.

$\displaystyle 2x-1 = -1$

Add one on both sides.

$\displaystyle 2x-1 +1= -1+1$

$\displaystyle 2x=0$

Divide by 2 on both sides.

$\displaystyle \frac{2x}{2}=\frac{0}{2}$

$\displaystyle x=0$

The answer is $\displaystyle 0$.

Rewrite the right side as base 2.

$\displaystyle \frac{1}{4} = 2^{-2}$

Replace the term into the equation.

$\displaystyle 2^{-3x+6} = 2^{-2}$

With similar bases, we can set the exponents equal.

$\displaystyle -3x+6 = -2$

Subtract six from both sides.

$\displaystyle -3x+6-6 = -2-6$

$\displaystyle -3x=-8$

Divide by negative three on both sides.

$\displaystyle \frac{-3x}{-3}=\frac{-8}{-3}$

The answer is:  $\displaystyle \frac{8}{3}$

Raise a fraction to a negative power by raising its reciprocal to the power of the absolute value of the exponent. Then apply the properties of exponents as follows:

$\displaystyle \left (8 x ^{3 } \right )^{-2} = \frac{1}{\left (8 x ^{3 } \right )^{ 2}} = \frac{1}{8^{ 2} \left ( x ^{3 } \right )^{ 2}} =\frac{1}{64 x ^{3 \cdot 2 } } =\frac{1}{64 x ^{6 } }$

To simplify this problem we need to find the least common denominator between the two fractions. To do this we look at 5 and at 8. The least common number between these two is 40.

In order to rewrite each fraction in terms of a denominator of 40 we need to muliple as follows:

$\displaystyle \frac{1}{5}*\frac{8}{8} + \frac{3}{8}* \frac{5}{5}$

we are able to mulitply by 8/8 and 5/5 because those fractions are really just 1 written in a different format.

Now using order of opperations we get the following

$\displaystyle \frac{8}{40}+\frac{15}{40}$

Now we have a common denominator and can do our addition to get the simplfied number:

$\displaystyle \frac{23}{40}$

In order to be able to find $\displaystyle x$, we must first find the least common denominator. In this case, it is $\displaystyle 14$: 

$\displaystyle \frac{2}{7}= \frac{2\times 2}{7\times 2}=\frac{4}{14}$

$\displaystyle \frac{1}{7}=\frac{1\times 2}{7\times 2}=\frac{2}{14}$

The equation can now be written as:

$\displaystyle \frac{3}{14} + \frac{4}{14} = x-\frac{2}{14}$

Solving for $\displaystyle x$, we get: 

$\displaystyle \frac{3}{14}+\frac{4}{14} + \frac{2}{14}=x$

$\displaystyle x=\frac{9}{14}$

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case $\displaystyle 2$ and $\displaystyle 4$ share a factor other than $\displaystyle 1$ which is $\displaystyle 2$. We can divide those numbers by $\displaystyle 2$ to get $\displaystyle 1$ and $\displaystyle 2$ leftover.

Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is $\displaystyle 4$.

$\displaystyle 2\cdot 4\rightarrow2(1\cdot 2)=4$

Another approach is to list out the factors of both number and find the factor that appears in both sets first.

$\displaystyle 2:2,4,6,8,10,...$

$\displaystyle 4:4,8,12,16,...$

We can see that $\displaystyle 4$ appears in both sets before any other number thus, this is our answer.

To simplify the sum of the two fractions, we must find the common denominator.

Simplifying the denominator of the first fraction, we get

$\displaystyle \frac{3}{x^2-1}=\frac{3}{(x-1)(x+1)}$

because the denominator is a difference of two squares, which follows the form

$\displaystyle (x^2-a^2)=(x+a)(x-a)$

Now, we can rewrite the sum as

$\displaystyle \frac{3}{(x+1)(x-1)}+\frac{4}{x+1}$

It is far easier to see the common denominator now:

$\displaystyle \frac{3}{(x-1)(x+1)}+\frac{4(x-1)}{(x-1)(x+1)}=\frac{3+4x-4}{(x-1)(x+1)}=\frac{4x-1}{x^2-1}$