Simplify each of the expressions to determine which satisfies the condition of the problem:

You should see on this problem that the numbers used, 2, 4, and 8, are all powers of 2. So to get the exponents in a way to be able to be used together, you can factor each base into a base of 2. That gives you:

Then you can apply the rule that when you take one exponent to another, you multiply the exponents. This then simplifies your equation to:

And now on the left hand side of the equation you can apply another exponent rule, that when you multiply two exponents of the same base, you add the exponents together:

Since the bases here are all the same, you can set the exponents equal. This gives you:

All of the exponent rules deal with multiplying, rather than adding, bases. In order to turn this into a multiplication question, we count apples (or chickens, or s, or whatever). How many ’s are there here? Three. This expression can be rewritten as . Now the exponent rules will apply; . The answer is .

All of the exponent rules deal with multiplying, rather than adding, bases. In order to turn this into a multiplication question, we count apples (or chickens, or s, or whatever). How many ’s are there here? Three. This expression can be rewritten as . Now the exponent rules will apply; . The answer is .

You should see on this problem that the numbers used, 2, 4, and 8, are all powers of 2. So to get the exponents in a way to be able to be used together, you can factor each base into a base of 2. That gives you:

Then you can apply the rule that when you take one exponent to another, you multiply the exponents. This then simplifies your equation to:

And now on the left hand side of the equation you can apply another exponent rule, that when you multiply two exponents of the same base, you add the exponents together:

Since the bases here are all the same, you can set the exponents equal. This gives you:

Simplify each of the expressions to determine which satisfies the condition of the problem:

This problem hinges on your ability to recognize 16, 4, and 64 all as powers of 4 (or of 2). If you make that recognition, you can use exponent rules to express the terms as powers of 4:

Since taking one exponent to another means that you multiply the exponents, you can simplify the numerator and have:

And then because when you divide exponents of the same base you can subtract the exponents, you can express this as:

This means that .

If you try to simplify the expression given in the question, you will have a hard time…it is already simplified! However, if you look at the four answer choices you will realize that most of these contain roots in the denominator. Whenever you see a root in the denominator, you should look to rationalize that denominator. This means that you will multiply the expression by one to get rid of the root.

Consider each answer choice as you attempt to simplify each.

For choice , the expression is already simplified and is not the same. At this point, your time is better spent simplifying those that need it to see if those simplified forms match.

For choice , employ the "multiply by one" strategy of multiplying by the same numerator as denominator to rationalize the root. If you do so, you will multiply by

, which is no the same as .

For answer choice , multiply by .

And since , you can simplify the fraction:

, which matches perfectly. Therefore, answer choice is correct.

NOTE: If you want to shortcut the algebra, this problem offers you that opportunity by leveraging the answer choices along with an estimate. You can estimate that the given expression, , is between and , because the is between (which is ) and (which is ). Therefore you know you are looking for a proper fraction, a fraction in which the numerator is smaller than the denominator. Well, look at your answer choices and you will see that only answer choice fits that description. So without even doing the math, you can rely on a quick estimate and know that you are correct.

Whenever you are asked to simplify an expression with exponents and two different bases, you should immediately look to factor. In this case, you should notice that both and are powers of . This means that you can rewrite them as and respectively.

Once you do this, the numerator becomes .

As you are raising an exponent to an exponent, you should then recognize that you need to multiply the two exponents in order to simplify to get .

Similarly, the denominator becomes , which you can simplify by multiplying the exponents to get .

Your fraction is therefore . Remember that to divide exponents of the same base, simply subtract the exponents. This gives you:

, or .

Whenever you are asked to simplify an expression with exponents and two different bases, you should immediately look to factor. In this case, you should notice that both and are powers of . This means that you can rewrite them as and respectively.

Once you do this, the numerator becomes .

As you are raising an exponent to an exponent, you should then recognize that you need to multiply the two exponents in order to simplify to get .

Similarly, the denominator becomes , which you can simplify by multiplying the exponents to get .

Your fraction is therefore . Remember that to divide exponents of the same base, simply subtract the exponents. This gives you:

, or .