Solving Problems with Roots & Exponents - ACT Math

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

,

To solve this problem we must first subtract a square from both sides

We move to the right 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

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 the 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.

The key to this problem is to avoid mistakes in finding with the root equation. There are a few different ways you could solve for :

1. Leverage the fact that and apply that to . That means that . Divide both sides by and see that , so .

2. Realize that (reverse engineering the root) and see that , so must equal .

However you find , you must then apply that value to the exponent expression in the second equation. Now you have . And since you're dealing with exponents, you will want to express as , meaning that you now have:

Here you should deal with the negative exponents, the rule for which is that . So the fraction you're given, , can then be transformed to .

Now you have:

Employing another rule of exponents, that of dividing exponents of the same base, you can transform the left-hand side to:

Since you now have everything with a base of , you can express as just . This then means that is the correct answer choice.

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