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

With similar bases, we can set the powers equal.

Add one on both sides.

Divide by 2 on both sides.

The answer is .

Rewrite the right side as base 2.

Replace the term into the equation.

With similar bases, we can set the exponents equal.

Subtract six from both sides.

Divide by negative three on both sides.

The answer is:  

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:

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:

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

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

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

The equation can now be written as:

Solving for , we get: 

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

In this case  and  share a factor other than  which is . We can divide those numbers by  to get  and  leftover.

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

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

We can see that  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

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

Now, we can rewrite the sum as

It is far easier to see the common denominator now:

To solve this equation, we must find the least common denominator to add the fractions.

Keep in mind that the denominator of the second term is the difference of squares, which can be rewritten as

This is the least common denominator.

Now, we multiply both sides of the equationby the LCD on top and bottom (this is essentially 1):

After canceling terms, we get

Now, we solve for x:

Find the least common denominator (LCD) and convert each fraction to the LCD and then add.  Simplify as necessary.

The result is an improper fraction because the numerator is larger than the denominator and can be simplified and converted to a mix numeral.

Start by making both fractions into the same denominator. One option is 

Then adjust the numerators by multiplying each fraction's numerator by the other fraction's denominator: 

Then add the adjusted numerators: 

Then we simplify by dividing both numerator and denominator by 2:

which gives us the final result.