SAT Mathematics : Solving Problems with Roots

Example Questions

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Example Question #1 : Solving Problems With Roots

Simplify

For what value of  is this equation true?

Explanation:

To solve for , we must first square both sides to get rid of the radical. We get
We subtract both sides by  to get the  alone.

We square root both sides to get

Answer choice  is incorrect because it was not square rooted.

Example Question #2 : Solving Problems With Roots

Simplify

For what value of  is this equation true?

Explanation:

To solve for , we must first square both sides to get rid of the radical. We get . We subtract both sides by  to get the  alone.

We divide by  to get  alone .

We square root both sides to get  Since  is not listed as an answer choice, we simplify. The highest square root that can multiply to  is . We take the  out of the radical to get .

Example Question #1 : Solving Problems With Roots

Simplify:

Explanation:

To solve this problem we must first simplify the radical by breaking it up into two parts  becomes   then we simplify into  to get

We multiply  to get , then divide by  to get

Example Question #5 : Solving Problems With Roots

Find the value of

Explanation:

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

Example Question #12 : Exponents & Roots

Find the value of

Explanation:

To solve this problem we must first subtract  from both sides

Then we square both sides

Divide both sides by

Example Question #7 : Solving Problems With Roots

Find the value of

Explanation:

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

Example Question #8 : Solving Problems With Roots

Find the value of

Explanation:

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

Example Question #8 : Solving Problems With Roots

Find the value of

and

and

Only

Only

and

Explanation:

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

Example Question #10 : Solving Problems With Roots

Which of the following is equivalent to ?

Explanation:

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.

Example Question #11 : Solving Problems With Roots

If  and , what is ?

Explanation:

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.

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