Factoring Common Factors of Squares and Square Roots - ACT Math

To solve the equation , we can first factor the numbers under the square roots.

When a factor appears twice, we can take it out of the square root.

Now the numbers can be added directly because the expressions under the square roots match.

First, we can simplify the radicals by factoring.

Now, we can factor out the .

Now divide and simplify.

To begin with, factor out the contents of the radicals. This will make answering much easier:

They both have a common factor . This means that you could rewrite your equation like this:

This is the same as:

These have a common . Therefore, factor that out:

These three roots all have a in common; therefore, you can rewrite them:

Now, this could be rewritten:

Now, note that

Therefore, you can simplify again:

Now, that looks messy! Still, if you look carefully, you see that all of your factors have ; therefore, factor that out:

This is the same as:

Begin by factoring out the relevant squared data:

is the same as

This can be simplified to:

Since your various factors contain square roots of , you can simplify:

Technically, you can factor out a :

Begin by breaking apart the square roots on the left side of the equation:

This can be rewritten:

You can combine like terms on the left side:

Solve by dividing both sides by :

This simplifies to:

To begin solving this problem, find the greatest common perfect square for all quantities under a radical.

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Pull out of each term on the left:

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Next, factor out from the left-hand side:

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Lastly, isolate :

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To begin solving this problem, find the greatest common perfect square for all quantities under a radical.

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Factor out the square root of each perfect square:

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Next, factor out from each term on the left-hand side of the equation:

--->

Lastly, isolate :

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Solving this one is tricky. At first glance, we have no common perfect square to work with. But since each term can produce the quantity , let's start there:

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Simplify the first term:

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Divide all terms by to simplify,

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Next, factor out from the left-hand side:

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Isolate by dividing by and simplifying:

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Last, simplify the denominator:

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Right away, we notice that is a prime radical, so no simplification is possible. Note, however, that both other radicals are divisible by .

Our first step then becomes simplifying the equation by dividing everything by :

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Next, factor out from the left-hand side:

--->

Lastly, isolate :

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Once again, there are no common perfect squares under the radical, but with some simplification, the equation can still be solved for :

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Simplify:

--->

Factor out from the left-hand side:

--->

Lastly, isolate :

--->

To begin solving this problem, find the greatest perfect square for all quantities under a radical. might seem intimidating, but remember that raising even single-digit numbers to the fourth power creates huge numbers. In this case, is divisible by , a perfect fourth power.

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Pull the perfect terms out of each term on the left:

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Next, factor out from the left-hand side:

--->

Lastly, isolate , remembering to simplify the fraction where possible:

--->

To start, begin pulling the largest perfect square you can out of each number:

So, . You can just add the two terms together once they have a common radical.

Again here, it is easiest to recognize that both of our terms are divisible by , a prime number likely to appear in our final answer:

Now, simplify our perfect squares:

Lastly, subtract our terms with a common radical:

To solve the equation , we can first factor the numbers under the square roots.

When a factor appears twice, we can take it out of the square root.

Now the numbers can be added directly because the expressions under the square roots match.

First, we can simplify the radicals by factoring.

Now, we can factor out the .

Now divide and simplify.

To begin with, factor out the contents of the radicals. This will make answering much easier:

They both have a common factor . This means that you could rewrite your equation like this:

This is the same as:

These have a common . Therefore, factor that out:

These three roots all have a in common; therefore, you can rewrite them:

Now, this could be rewritten:

Now, note that

Therefore, you can simplify again:

Now, that looks messy! Still, if you look carefully, you see that all of your factors have ; therefore, factor that out:

This is the same as:

Begin by factoring out the relevant squared data:

is the same as

This can be simplified to:

Since your various factors contain square roots of , you can simplify:

Technically, you can factor out a :

Begin by breaking apart the square roots on the left side of the equation:

This can be rewritten:

You can combine like terms on the left side:

Solve by dividing both sides by :

This simplifies to: