Exponents & Roots - ACT Math

Whenever you exponential expressions in both the numerator and denominator of a fraction, your first inclination might be to quickly simplify the expressions by canceling terms out in the numerator and denominator.

However, remember to follow the order of operations: you must simplify the numerator and denominator separately to revolve the exponents raised to exponents problem before you can look to cancel terms in the numerator and denominator.

If you consider the numerator, , you should recognize that, because there is no addition or subtraction within the parentheses, that you can simplify this expression by multiplying each exponent within the parentheses by to get:

.

Similarly, you can simplify the denominator, by multiplying each exponent within the parentheses by to get:

.

You can then recombine the numerator and denominator to get . Notice that you now have simple division. Remember that anytime you want to combine two exponential expressions with a common base that are being divided, you simply need to subtract the exponents. If you do, you get:

, which simplifies to .

Remember that if you forgot any of the rules, such as how to combine exponents with common bases that are being divided or whether you add or multiply when raising a power to a power, that you can always remind yourself how the rules work by testing small numbers.

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 see addition or subtraction with algebraic terms, you should only think about combining like terms or factoring. Here you have two of one term and three of another term so:

The difficulty in this problem relates primarily to common mistakes with factoring and exponent rules. If you understand exponent rules and how to combine like terms, you will answer this problem quickly and confidently. You should note that the answer choices do not really help you here – they are traps if you make a mistake with exponent rules! Many algebra problems on the ACT exploit common mistakes relating to certain content areas. For instance, in this example, you can see how easy it would be to accidentally add the exponents or add the bases. If you ever make one of these common mistakes, take note and be sure to avoid it the next time you see a similar problem.

This problem tests your ability to combine exponents algebraically, using both the distributive property and the rule for multiplying exponents with the same base. Here it is also helpful to look at the answers to see what the test maker is looking for. In the answer choices, the maximum number of individual terms is 2, and all terms involve a base 10 (or 100). So you should see that your goal is to rewrite as much as possible of what you're given in terms of 10.

In terms of factors/multiples, a 10 is created any time you can pair a 2 with a 5. So as you take the given expression:

Recognize that if you can pair the two 2s you have with two of the 5s in 5555, you'll be able to consider them 10s. So you can rewrite the given expression as:

From there, you can combine the and terms:

And since those are two bases, multiplied, each taken to the same exponent, they'll combine to. That can be rewritten as , making your expression:

From here, you'll apply the rule that and add the exponents from the 10s. That gives you:

The key to beginning this problem is finding common bases. Since 2, 4, and 8 can all be expressed as powers of 2, you will want to factor the 4 into and the 8 into so that can be rewritten as .

From there, you will employ two core exponent rules. First, when you take an exponent to another, you'll multiply the exponents.

That means that:

becomes

and

becomes

So your new expression is .

Then, when you're multiplying exponents of the same base, you add the exponents. So you can sum 2 + 8 + 24 to get 34, making your simplified exponent .

With this exponent problem, the key to getting the given expression in actionable form is to find common bases. Since both 9 and 27 are powers of 3, you can rewrite the given expression as:

When you've done that, you're ready to apply core exponent rules. When you take one exponent to another, you multiply the exponents. So for your numerator:

becomes . And for your denominator: becomes . So your new fraction is:

Next deal with the negative exponents, which means that you'll flip each term over the fraction bar and make the exponent positive. This then makes your fraction:

From there, recognize that when you divide exponents of the same base, you subtract the exponents. This means that you have:

which simplifies to , so your answer is .

This problem rewards those who see that roots and exponents are the same operations (roots are "fractional exponents"), and who therefore choose the easier order in which to perform the calculation. The trap here is to have you try to square 27. Not only is that labor-intensive, but once you get to 729 you then have to figure out how to take the cube root of that!

Because you can handle the root and the exponent in either order (were you to express this as a fractional exponent, it would be , which proves that the root and exponent are the same operations), you can take the cube root of 27 first if you want to, which you should know is 3. So at that point, your problem is what is ?" And you, of course, know the answer: it's 9.

It is important to be able to convert between root notation and exponent notation. The third root of a number (for example, is the same thing as taking that number to the one-third power .

So when you see that you're taking the third root of , you can read that as to the power:

This then allows you to apply the rule that when you take one exponent to another power, you multiply the powers:

This then means that you can express this as:

With roots, it is important that you are comfortable with factoring and with expressing roots as fractional exponents. A square root, for example, can be expressed as taking that base to the power. Using that rule, the given expression, , could be expressed using fractional exponents as:

This would allow you to then add the exponents and arrive at:

Since that 2 in the denominator of the exponent translates to "square root," you would have the square root of :

If you were, instead, to work backward from the answer choices, you would see that answer choice factors to the given expression. If you start with:

You can express that as:

That in turn will factor to:

The first root then simplifies to , leaving you with:

Therefore, as you can see, choice factors directly back to the given expression.

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

This problem tests your fluency with exponent rules, and gives you a helpful clue to guide you through using them. Here you may see that both 27 and 9 are powers of 3. and . This allows you to express as and as . Then you can simplify those exponents to get . Since when you divide exponents of the same base you subtract the exponents, you now have , and since you really have:

This then tells you that .

Note that had you not immediately seen to express all the numbers in this problem as powers of 3, the fact that the question asks for such a combination of variables, , should be your clue; you're given an exponent problem and asked for a subtraction answer, so that should get you thinking about dividing exponents of the same base to subtract the exponents, and at least give you some fodder for playing with exponent rules until you find a way to make progress.

An important principle of exponents being tested here is that when you multiply/divide exponents of the same base, you add/subtract those exponents. Here you can do the corollary; if you had , you would add together those exponents to get . But in this case you're given the combined exponent and may want to convert it to so that you can factor:

allows you to factor the terms to get:

You can do the arithmetic to simplify , allowing you to then divide both sides by 3 and have:

So .

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 .

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:

This problem rewards your ability to factor exponents. Here if you factor out common terms in the given equation, you can start to see how the math looks like the correct answer. Factoring negative exponents may feel a bit different from the more traditional factoring that you do more frequently, but the mechanics are the same. Here you can choose to factor out the biggest "number" by sight, , or the number that's technically greatest, . Because all numbers are 2-to-a-power, you'll be factoring out common multiples either way.

If you factor the common , the expression becomes:

Here you can do the arithmetic on the smaller exponents. They convert to:

When you sum the fractions (and 1) within the parentheses, you get:

And since you can express this now as:

, which converts to the correct answer:

Note that you could also have started by factoring out from the given expression. Had you gone that route, the factorization would have led to:

This also gives you the correct answer, as when you sum the terms within parentheses you end up with:

Whenever you are given addition or subtraction of two exponential terms with a common base, a good first instinct is to factor the addition or subtraction problem to create multiplication. Most exponent rules deal with multiplication/division and very few deal with addition/subtraction, so if you're stuck on an exponent problem, factoring can be your best friend.

For the equation , can be rewritten as , leveraging the rule that when you multiply exponents of the same base, you add the exponents. This allows you to factor the common term on the left hand side of the equation to yield:

And of course you can simplify the small subtraction problem within parentheses to get:

And you can take even one further step: since everything in the equation is an exponent but that 4, you can express 4 as to get all the terms to look alike:

Now you need to see that can be expressed as or as . So the equation can look like:

You can then divide both sides by and be left with:

This proves that .

To solve this problem we must recognize that can be broken down into

After breaking into we see that the cancel out

This leaves us with

To solve this problem we must first simplify into and further into

Then we can multiply to get

To find we first cancel out the on both sides and then divide by and get

To solve this problem we must first subtract from both sides

Then we square both sides

Add the to both sides

Divide both sides by

To solve this problem we first multiply both sides by to get rid of the fraction

Then we add to both sides

We move to the left side to set the equation equal to . This way we are able to factor the equation as if it was a quadratic.

And now we can factor into

Therefore the value of is

, does not exist

,