Re: Simplification Of Radicals Pdf Download
Emmanuelle Riker
Radicals were introduced in previous tutorial when we discussed real numbers. For example, root(25) = 5, and root(2) = 1.4142135 ... (an infinite nonrepeating decimal). We are now interested in developing techniques that will aid in simplifying radicals and expressions that contain radicals. In this text, we will deal only with radicals that are square roots. Other radicals, such as cube roots and fourth roots , will be discussed in later algebra courses.
The following two properties of radicals are basic to the discussion.
While it's a simple consequence of the distributive law for exponents, one can view it from a more general perspective. Namely, both $\rm\ \sqrt3/2\ $ and $\rm\ \sqrt3/\sqrt2\ $ are roots of $\rm\ x^2 - 3/2 $. But this polynomial has a unique positive root since it is increasing from $\rm\: -3/2\ $ to $\: +\infty\ $ on $\ [0,\infty)\:.\ $ Therefore this uniqueness theorem implies that the two roots are equal. One can frequently apply analogous techniques to more much complicated expressions involving messy nested radicals - inferences far removed from the laws of exponents. Thus I point out once again what I have emphasized in many varied posts here: uniqueness theorems provide powerful tools for proving equalities.
I think it would be interesting to try this out. Some students may prefer this method, but most students will likely move towards simplifying radicals without drawing pictures. But by drawing pictures as they are learning this skill, students will be connecting mathematical ideas and building conceptual understanding. New learning (simplifying radicals in Math 10) will be connected to prior learning (concept of a square root introduced in Math 8). Students will have a more solid understanding of why perfect squares are used.
If we mindlessly apply the radical simplification idea to anything that "doesn't produce", we run the risk of turning higher education into a corporate dystopia where we are all slaves to shareholder capitalism. Regular readers of this blog know that I am on record as saying higher education has a lot to learn from the corporate sector, and we would be well served to be curious, not judgmental about the corporate world, learning where we can and appropriating what can be useful. But a slavish devotion to efficiency and "shareholder value" is not one of those things. (And if you look around, the best companies out there don't subscribe to that devotion themselves.)
When we encounter a potential simplification that incurs a human cost, or a cost to the core mission of the institution, we need to take a breath and think carefully about what "serving a purpose" means. If you're a small liberal arts college, for example, and the number of majors in your Philosophy program is declining, the shareholder-capitalist response would say: Downsize the Philosophy Department and reinvest the resources in something that's "working". Some schools are doing exactly that. This makes sense from the standpoint of maximizing "shareholder" value. But it doesn't make sense from the standpoint of being a liberal arts college. At least, cutting programs and faculty should be the response of last resort, once a good-faith effort to build that program back up has taken place.
Simplifying radicals is the process of manipulating a radical expression into a simpler or alternate form. Generally speaking, it is the process of simplifying expressions applied to radicals.
This means that when we are dealing with radicals with different radicands, like \(\sqrt5\) and \(\sqrt7\), there is really no way to combine or simplify them. However, when dealing with radicals that share a base, we can simplify them by combining like terms.
When multiplying radicals, we make extensive use of the identity \(\sqrtab=\sqrta\times\sqrtb\). This means that two radicals, when multiplied together, might produce an integer rather than another radical.
Fractions are not considered to be written in simplest form if they have an irrational number \(\big(\)like \(\sqrt2\), for example\(\big)\) in the denominator. We can simplify the fraction by rationalizing the denominator. This is a procedure that frequently appears in problems involving radicals.
When we have a more complicated expression involving a radical, we must look for another approach. However, when the denominator is a binomial expression involving radicals, we can use the difference of two squares identity to produce a conjugate pair that will remove the radicals from the denominator. For example, if we want to remove the radicals from the expression \(\sqrt2+\sqrt3\), we can multiply it by its conjugate pair \(\sqrt2 - \sqrt3\).
I'm in a self paced College Algebra course and I just got to simplifying radicals. It is kicking my backside. Come to find out it's a part of "beginning algebra" which means you're supposed to know it before college algebra. Of all the math I've done so far in the course I find this the most esoteric.
In this tutorial we will talk about rationalizing the denominator andnumerator of rational expressions. Recall from Tutorial3: Sets of Numbers that a rational number is a number that canbe written as one integer over another. Recall from Tutorial3: Sets of Numbers that an irrational number is not one that ishard to reason with but is a number that cannot be written as one integerover another. It is a non-repeating, non-terminating decimal. Oneexample of an irrational number is when you have a root of an expressionthat is not a perfect root, for example, the square root of 7 or the cuberoot of 2. So when we rationalize either the denominator or numeratorwe want to rid it of radicals.
Step 2: Make sure all radicals are simplified AND Step 3: Simplify the fraction if needed.
*Cube root of 27 a cube is 3a
As discussed in example 1, we would not be able to cancel out the 3 with the 18 in our final fraction because the 3 is on the outside of the radical and the 18 is on the inside of the radical. Also, we cannot take the cube root of anything under the radical. So, the answer we have is as simplified as we can get it.
Rationalizing the Numerator
(with one term)
As mentioned above, when a radical cannot be evaluated, for example, the square root of 3 or cube root of 5, it is called an irrational number. So, in order to rationalize the numerator, we need to get rid of all radicals that are in the numerator. Note that these are the same basic steps for rationalizing a denominator, we are just applying to the numerator now.
Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the numerator.
If the radical in the numerator is a square root, then you multiply by a square root that will give you a perfect square under the radical when multiplied by the numerator. If the radical in the numerator is a cube root, then you multiply by a cube root that will give you a perfect cube under the radical when multiplied by the numerator and so forth... Note that the phrase "perfect square" means that you can take the square root of it. Just as "perfect cube" means we can take the cube root of the number, and so forth. Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent.
Step 2: Make sure all radicals are simplified.
Some radicals will already be in a simplified form, but make sure you simplify the ones that are not. If you need a review on this, go to Tutorial 39: Simplifying Radical Expressions.
Step 3: Simplify the fraction if needed.
Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.
Example 3: Rationalize the numerator .
Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the numerator.
Since we have a square root in the numerator, then we need to multiply by the square root of an expression that will give us a perfect square under the radical in the numerator. So in this case, we can accomplish this by multiplying top and bottom by the square root of 5:
*Mult. num. and den. by sq. root of 5
*Num. now has a perfect square under sq. root
Step 2: Make sure all radicals are simplified. AND Step 3: Simplify the fraction if needed.
*Sq. root of 25 is 5
As discussed above, we would not be able to cancel out the 5 with the 30 in our final fraction because the 5 is on the outside of the radical and the 30 is on the inside of the radical. Also, we cannot take the square root of anything under the radical. So, the answer we have is as simplified as we can get it.
Example 4: Rationalize the numerator .
Step 1: Multiply numerator and denominator by a radical that will get rid of the radical in the numerator.
Since we have a cube root in the numerator, we need to multiply by the cube root of an expression that will give us a perfect cube under the radical in the numerator. So in this case, we can accomplish this by multiplying top and bottom by the cube root of :
*Mult. num. and den. by cube root of
*Num. now has a perfect cube under cube root
Step 2: Make sure all radicals are simplified AND Step 3: Simplify the fraction if needed.
*Cube root of 8 x cube is 2x
As discussed above, we would not be able to cancel out the 2x with the 4 x squared in our final fraction, because the 2x is on the outside of the radical and the 4 x squared is on the inside of the radical. Also, we cannot take the cube root of anything under the radical. So, the answer we have is as simplified as we can get it.
Rationalizing the Denominator
(with two terms)
Above we talked about rationalizing the denominator with one term. Again, rationalizing the denominator means to get rid of any radicals in the denominator. Because we now have two terms, we are going to have to approach it differently than when we had one term, but the goal is still the same.
Step 1: Find the conjugate of the denominator.
You find the conjugate of a binomial by changing the sign that is between the two terms, but keep the same order of the terms. a + b and a - b are conjugates of each other.
Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1 .
Keep in mind that as long as you multiply the numerator and denominator by the exact same thing, the fractions will be equivalent.
Step 3: Make sure all radicals are simplified.
Some radicals will already be in a simplified form, but make sure you simplify the ones that are not. If you need a review on this, go to Tutorial 39: Simplifying Radical Expressions.
Step 4: Simplify the fraction if needed.
Be careful. You cannot cancel out a factor that is on the outside of a radical with one that is on the inside of the radical. In order to cancel out common factors, they have to be both inside the same radical or be both outside the radical.
Example 5: Rationalize the denominator
Step 1: Find the conjugate of the denominator.
In general the conjugate of a + b is a - b and vice versa. So what would the conjugate of our denominator be? It looks like the conjugate is .
Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1.
*Mult. num. and den. by conjugate of den.
*Product of the sum and diff. of two terms
Step 3: Make sure all radicals are simplified AND Step 4: Simplify the fraction if needed.
No simplifying can be done on this problem so the final answer is:
Example 6: Rationalize the denominator .
Step 1: Find the conjugate of the denominator.
In general the conjugate of a + b is a - b and vice versa. So what would the conjugate of our denominator be? It looks like the conjugate is .
Step 2: Multiply the numerator and the denominator of the fraction by the conjugate found in Step 1 .
*Mult. num. and den. by conjugate of den.
*Use FOIL method in num.
*Product of the sum and diff. of two terms
Step 3: Make sure all radicals are simplified AND Step 4: Simplify the fraction if needed.
*12 is (4)(3) and sq. root of 4 is 2
*18 is (9)(2) and sq. root of 9 is 3