Start by simplifying the numerator and denominator separately. In the numerator, (c³)² is equal to c⁶. In the denominator, c² * c⁴ equals c⁶ as well. Dividing the numerator by the denominator, c⁶/c⁶, gives an answer of 1, because the numerator and the denominator are the equivalent.

The numerator is simplified to  (by adding the exponents), then cube the result. a²⁴/a⁶ can then be simplified to .

The easiest way to solve this is to simplify the fraction as much as possible. We can do this by factoring out the greatest common factor of the numerator and the denominator. In this case, the GCF is . 

Now, we can cancel out the  from the numerator and denominator and continue simplifying the expression.

(300)(400) = 120,000 or 12 * 10⁴.

The three two multiply to become 8 and the powers of ten can be added to become 10²¹.

 and  can be multiplied together to give you  which is the first part of our answer. When you multiply exponents with the same base (in this case, ), you add the exponents. In this case,  should give us  because . The answer is 

1. Solve for x in 3^x = 27. x = 3 because 3 * 3 * 3 = 27.
2. Since x = 3, one can substitute x for 3 in 2^2x 
3. Now, the expression is 2^2*3
4. This expression can be interpreted as 2^{2 *} 2² * 2². Since 2² = 4, the expression can be simplified to become 4 * 4 * 4 = 64.
5. You can also multiply the powers to simplify the expression. When you multiply the powers, you get 2⁶, or 2 * 2 * 2 * 2 * 2 * 2
6. 2⁶ = 64.

In order to solve this equation, we first need to find a common base for the exponents. We know that 2³ = 8 and 2⁴ = 16, so it makes sense to use 2 as a common base, and then rewrite each side of the equation as a power of 2.

8^x-3 = (2³)^x-3

We need to remember our property of exponents which says that (a^b)^c = a^bc.

Thus (2³)^x-3 = 2^3(x-3) = 2^{3x - 9}.

We can do the same thing with 16^4-x.

16^4-x = (2⁴)^4-x = 2^4(4-x) = 2^16-4x.

So now our equation becomes

2^{3x - 9} = 2^16-4x

In order to solve this equation, the exponents have to be equal. 

3x - 9 = 16 - 4x

Add 4x to both sides.

7x - 9 = 16

Add 9 to both sides.

7x = 25

Divide by 7.

x = 25/7.

We can start by rewriting 4¹¹ as 4 * 4¹⁰. This will allow us to collect the like terms 4¹⁰ into a single term.

4¹⁰ + 4¹⁰ + 4¹⁰ + 4¹⁰ + 4¹¹

= 4¹⁰ + 4¹⁰ + 4¹⁰ + 4¹⁰ + 4 * 4¹⁰

= 8 * 4¹⁰

Because the answer choices are written with a base of 2, we need to rewrite 8 and 4 using bases of two. Remember that 8 = 2³, and 4 = 2².

8 * 4¹⁰

= (2³)(2²)¹⁰

We also need to use the property of exponents that (a^b)^c = a^bc. We can rewrite (2²)¹⁰ as 2^2x10 = 2²⁰.

(2³)(2²)¹⁰

= (2³)(2²⁰)

Finally, we must use the property of exponents that a^b * a^c = a^b+c.

(2³)(2²⁰) = 2²³

The answer is 2²³.

3 + 3ⁿ⁺³ = 81

In this equation, there is a common factor of 3, which can be factored out.

Thus, 3(1 + 3ⁿ⁺²) = 81

Note: when 3 is factored out of 3ⁿ⁺³, the result is 3ⁿ⁺² because (3ⁿ⁺³ = 3¹ * 3ⁿ⁺²). Remember that exponents are added when common bases are multiplied.  Also remember that 3 = 3¹.

3(1 + 3ⁿ⁺²) = 81

(1 + 3ⁿ⁺²) = 27

3ⁿ⁺² = 26

Note: do not solve for n individually.  But rather seek to solve what the problem asks for, namely 3ⁿ⁺².