First, reduce all values to a common base using properties of exponents.

Plugging back into the equation-

Using the formula

We can reduce our equation to

So,

Using the rules of exponents, 23 + 22 = 8 + 4 = 12

When you multiply exponents, you add the common bases:

y4 x7 + y5x6 + y18x4 + y3x26

When multiplying bases that have exponents, simply add the exponents. Note that you can only add the exponents if the bases are the same. Thus:

Since the base is 5 for each term, we can say 2 + n =12. Solve the equation for n by subtracting 2 from both sides to get n = 10.

Rewrite the term on the left as a product. Remember that negative exponents shift their position in a fraction (denominator to numerator).

The term on the right can be rewritten, as 27 is equal to 3 to the third power.

Exponent rules dictate that multiplying terms allows us to add their exponents, while one term raised to another allows us to multiply exponents.

We now know that the exponents must be equal, and can solve for .

The only value of x where the two equations equal each other is 1. All you have to do is substitute the answer choices in for x.

The rule for adding exponents is . We can thus see that and are no more compatible for addition than and are.

You could combine the first two terms into , but note that PEMDAS prevents us from equating this to (the exponent must solve before the distribution).

Since the problem requires us to finish in a power of 2, it's easiest to begin by reducing all terms to powers of 2. Fortunately, we do not need to use logarithms to do so here.

Thus,

h–2n = 1/h2n

hn + h–2n = hn + 1/h2n