Fundamental Properties of Roots & Exponents - 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:

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: