How to divide square roots
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GRE Quantitative Reasoning › How to divide square roots
Solve for :
Explanation
To get rid of the radical, multiply top and bottom by .
Square both sides.
Simplify:
Explanation
Let's combine the two radicals into one radical and simplify.
Remember, when dividing exponents of same base, just subtract the power.
The final answer is .
Simplify.
Explanation
To get rid of the radical, we need to multiply by the conjugate. The conjugate uses the opposite sign and multiplying by it will let us rationalize the denominator in this problem. The goal is getting an expression of in which we are taking the differences of two squares.
Rationalize the denominator:
Explanation
We don't want to have radicals in the denominator. To get rid of radicals, just multiply top and bottom by that radical.
Simplify:
Explanation
Let's factor the square roots.
Then, multiply the numerator and the denominator by to get rid of the radical in the denominator.
Rationalize the denominator and simplify:
Explanation
We don't want to have radicals in the denominator. To get rid of radicals, just multiply the numerator and the denominator by that radical.
Remember to distribute the radical in the numerator when multiplying.
This may be the answer; however, the numerator can be simplified. Let's factor out the squares.
Finally, if we factor out a , we get:
Which of the following is equal to
Explanation
We then multiply our fraction by
because we cannot leave a radical in the denominator. This gives us
. Finally, we can simplify our fraction, dividing out a 3, leaving us with
Solve for :
Explanation
If we multiplied top and bottom by , we would get nowhere, as this would result:
. Instead, let's cross-multiply.
Then, square both sides to get rid of the radical.
Divide both sides by .
The reason the negative is not an answer is because a negative value in a radical is an imaginary number.
Which of the following is equivalent to ?
Explanation
We can definitely eliminate some answer choices. and
don't make sense because we have an irrational number. Next, let's multiply the numerator and denominator of
by
. When we simplify radical fractions, we try to eliminate radicals, but here, we are going to go backwards.
, so
is the answer.
Simplify:
Explanation
Let's get rid of the radicals in the denominator of each individual fraction.
Then find the least common denominator of the fractions, which is , and multiply them so that they each have a denominator of
.
We can definitely simplify the numerator in the right fraction by factoring out a perfect square of .
Finally, we can factor out a :
That's the final answer.