Rationalizing Denominators Composed by Trinomials.
1st. We place 2 square roots of the denominator in parenthesis:
Then, we write the remaining term with its corresponding sign:
Actually, its the same as:, but it is easier for us to know the conjugate of the denominator: .
Be careful now that the first term is composed by two addends (placed in parenthesis).
2nd. We multiply the numerator and denominator by the conjugate of the denominator. Be careful with the parenthesis:
We have to bear in mind that the first term of the denominator is composed by two addends and when we multiply the sum of two numbers by their difference, we will get the difference of their squares.
Thus, the square of the first term is written as a sum elevated to the power of 2: .
We develop the square of the first term of the denominator:
3rd. We have to multiply both members of the fraction again by the conjugate of the denominator:
Simplify the numerator and denominator by 2:
To simplify a product of several factors by a number, we just have to simplify ONE factor.
We can continue simplifying the numerator and denominator. To do this, I must calculate the common factor of 2 in :
Finding the Square Root of a Polynomial.
This is not an operation performed commonly. However, it is a good idea to know the procedure of extracting the square root of a Polynomial.
Finding the square root of a monomial is a very simple task:
We simply need to find the square root of each factor: the numeric part (if it has any), and the letter variable part by dividing the exponent of each factor by 2, because we are dealing with a square root.
10.91 Find the square root of:
We extract everything we can from each factor. Since 48 is equal to 16 x 3, we can bring out 4 as a square root of 16 and keeping 3 inside the square root.
Whenever the exponent of the sub-radical quantity is larger than the index of a radical, we divide the exponent by the index. The quotient indicates the exponent of the factor outside the square root and its remainder is the exponent of that factor left inside the square root:
10.92 Find the square root of:
Let's find the square root of a Polynomial:
1st. Everything needs to be in alphabetical order (in this case, the radicant is ordered from 'x' to 'y') and in a descendant order:
2nd. We calculate the square root of the first term of the Polynomial and we write the result in its corresponding location:
3rd. We raise the result to the square and we write it in the radicant directly below the first term changing its sign. We perform the operation and we lower the following terms (two remaining terms in the radicant):
4th. We multiply the square root calculated up until now by 2 and we write it below:
5th. We divide the first term we now have in the radicant by the square root times 2 we just wrote. This result, with its corresponding sign, will be the second term in the result:
6th. We write the last calculated result next to the square root times 2 we calculated previously:
7th. The two terms composed by: the square root I calculated first times two and the second term in the result are placed in parenthesis and multiply them by this last term:
8th. I perform the operation (multiplication) and I change the sign of the product and place it below the two terms I have in the radicant (the terms I lowered in step 3):
In case we have more terms in the radicant:
we would lower any necessary terms to complete three terms;
we would multiply the square root calculated up to that moment times 2 placing the result in a new second line (under the square root multiplied by two we found previously);
we would divide the first term of the rest (radicant) by the first term of the value calculated in the step we did before. This value would be the third term in our answer. As you can see, we are repeating the steps we have explained previously.
10.93 Calculate the square root of:
After checking if the polynomial in the radicant is ordered (from 'x' to 'y' and descendant exponents), we find that the third and fourth terms are not in order. We correct their position. Here you have the full square root following the steps we followed in the exercise above:
10.94 Calculate the square root of:
First, check if the polynomial is properly ordered.
Place similar terms properly.
This is the square root solved step-by-step:
10.95 Calculate the square root of: