Rationalization of simple surd
Simplification And Rationalization Of Simple Surds
Surds are numbers left in root form (√) to express its precise value. It possesses an infinite number of non-repeating decimals. Thus, surds are irrational numbers.
There are specific rules that we follow to simplify an expression involving surds. Rationalization of the denominator is a way to simplify these expressions. It is performed by eliminating the surd in the denominator. This is illustrated in Rules 3, 5 and 6.
It can frequently be essential to obtain the biggest perfect square factor in order to simplify surds. The largest perfect square factor is obtained by looking at any possible factors of the number that is being square rooted. For an example, if you are looking at the square root of 242. Can you simplify this? But 2 x 121 is 242 and we can use the square root of 121 without leaving a surd (because we get 11). Since we cannot take the square root of a larger number that can be multiplied by another to give 242 then we say that 121 is the largest perfect square factor.
Six Rules of Surds
Rule 1:
An Example:
Simplify
Because 18=9× 2=32 ×2,since 9 is the biggest t perfect square factor of 18.
Rule 2:
An Example:
simplify:
Rule 3:
By multiplying both the numerator and denominator by the denominator you can rationalize the denominator.
Example:
Rationalize
:
Rule 4:
Example:
Simplify :
Rule 5:
Following this rule allows you to rationalize the denominator.
Example:
Rationalize :
Rule 6:
This rule allows you to rationalize the denominator.
Example:
Rationalize :
In elementary algebra, root rationalizationis a process by which surds in the denominator of an irrational fraction are removed.
These surds may be monomials or binomials and involves square roots, in simple examples. There are broad extensions to the technique.
At this level students can rationalise expressions with surds in the denominator as in Example 1a below.
Before level 10 students can simplify surds as in Example 1b below and add, subtract and multiply expressions that is composed of surds without the use of technology.
More advanced students will be asked to rationalize expressions like the one in Example 1c below.
Rationalization of surds make use of the students knowledge of fraction
You would find it difficult to rationalize surds if you have problems multiplying and simplifying fractions.
You are required to understand that multiplication by one, in a lot of forms (like in Example 1d below) results in an equivalent surd (i.e. the value is not altered by this process).
Rationalization of more complex surds
You would be asked to rationalize surds like the one in Example 2a below, at first but at a more advanced level, you will be required to rationalize more complex surds like Example 2b. If students just focus on section of the denominator when rationalizing Example 2b , they may multiply by 1 or -1, disguised as in Example 2c or Example 2d, abandoning a surd term that is left in the denominator.
Strategies that would help you to understand rationalization
Below are a few activities intended to make available a motivation for the process of rationalization and exploration to illustrate the equivalence of the rationalized and un-rationalized expressions. Students can check their exact calculations at all stages with approximate calculator calculations.
The activities are:
• Rationalization
• Complex surd Rationalization
• Golden rectangle
Rationalization makes available an exploration, a motivation and practice at rationalization of simple surds.
Complex surds rationalization makes available an exploration and practice for more complex surds, needing good fraction and algebra skills. Golden rectangle makes available a context for practicing calculations with surds. Calculating side lengths needs addition and subtraction, calculating areas needs multiplication and, for advanced students, illustrating that all the rectangles are the same shape requires division and rationalizing complex surd expressions.
Rationalization
This activity is made up of three components where students gradually build up to rationalize complex surds.
First component:
Identify one as a fraction in a lot of disguises
This short introduction swiftly reviews fractions equal to one, extending beyond familiar fractions. To learn this student are normally divided in pairs to identify which of the following are equal to one and provide the explanation to why it is so. Students could as well be asked to provide a few other fractions equal to one and made up of at least one surd.
| 3/3 | -3/3 | x/x |
| 1-2x/1-2x | 1-d/1+d | |
The second component :
Produce equivalent fractions
Start with any simple surd and multiply it by a fraction equal to 1 many times, obtaining a sequence of expressions, as illustrated below. Students can deduce that these will all have equivalent value, and calculate them on the calculator to counter-check. In the case below the value is 0.707…You can repeat this with other simple surd expressions.
The Third component:
Make available a motivation for rationalization
Rationalization involves rewriting a surd so that the denominator does not contain a surd term. In this procedure an equivalent surd (one with equivalent value) is created. Observe the similarity with fraction terminology.
At this stage, students are shown why rationalization was essential in the past. Try and calculate Example 5a below, by dividing 1 by the square root of 2 by hand, and calculate Example 5c by dividing the square root of 2 by 2. Compare the simplicity of calculating the rationalized expression over the un-rationalized expression. They both are made up of the same numerical value of 0.707…
Rationalization of surds is one of a lot of conventions of good mathematical writing style, therefore textbooks and computer algebra technology will usually rationalize surds before offering answers.
Broaden the range of expressions
At this stage, you would be shown how to rationalize an expression like that in Example 6a, emphasizing that the purpose is to rewrite this number in order that the denominator is not made up of a surd term. Revisit the idea that multiplying by 1 (in the form of Example 6b) gives rise to the same number, but in the form of Example 6c.
You may be asked to say what fraction that you would use to multiply surds like those illustrated below to be able to rationalize the surds.
You may for example in the second expression students may think that they ought to multiply top and bottom by 2 times the square root of 3, which would provide the correct result, but requires an extra ‘cancelling’ step to obtain the simplest form.
Rationalization of a monomial square root and cube root
For the fundamental procedure, the numerator and denominator ought to be multiplied by the same factor.
Example 1:
To rationalize this radical, bring in the factor :
The cube root disappears from the denominator, due to the fact that it is cubed:
This provides the result, after simplification:
Dealing with more square roots
For a denominator that is:
Rationalization can be obtained by multiplying by the Conjugate:
and applying the difference of two squares identity, which in this situation will give rise to −1. To obtain this result, the whole fraction ought to be multiplied by
This technique functions much more generally. It can readily be adapted to remove one square root at a time, i.e. to rationalize
by multiplication by
Example:
The fraction ought be multiplied by a quotient made up of .
Now, we can move on to remove the square roots in the denominator:
Generalizations
Rationalization can be extended to every algebraic numbers and algebraic functions (as an application of norm forms). For instance, to rationalize a cube root, two linear factors that involves cube roots of unity ought to be used, or in the same way a quadratic factor.
Surds, indices and logarithms are closely linked. They are, the majority of the time, studied together. Therefore, the exposure to indices and logarithms in our earlier tutorials will assist you to comprehend the use of surds. Numbers whose square roots cannot be determined in terms of rational numbers e.g,.√2,√3,√5 etc. are known as surds. Such numbers exist frequently in Trigonometry when calculating the ratio of angles; e.g, Cos 30=√3/2, tan 60=√3,tan30=1/√3; and in coordinate geometry in the calculation of distances. You will thus find it essential to have a sound knowledge of surds.
You ought to as well be able to recall from your previous knowledge of numbers that a number that is "square" is one that can be expressed as the square of a few other rational number. For instance
• 9=32
• 81=92
• (4/9)=(2/3)2
But not all numbers are rational numbers that is they do not have exact square roots. Examples are
• √2
• √3
• √5
• √8
• 2√3, etc
The square roots of numbers that do not have exact square roots are called Surds.
Such numbers are referred to as irrational numbers.
Notwithstanding that the approximate square roots of such numbers (irrational) can be obtained from tables of square roots, it is normally simpler to work with SURDS themselves. Observe as well that at this level whenever you deal with square roots, only positive square roots are considered.
Reduction to Basic Form
Any surd which is made up of a square number as a factor within the radical (i.e, the square root sign) is not in the basic form. For instance
• √2
• √50
• √108
are not in the basic form since they can be reduced further through simplification.
The examples below will demonstrate this concept: '√27=√(9X3)=''''√9√3=3√3 1.2 √50=√(25 X 2)=√25√2=5√2 1.3√108=√(36 X 3)=√36√3=6√3 From(1.1), (1.2) and 1.3, 3√3,5√2 and 6√3 cannot be simplified further; they have been reduced to the basic form. Surds that cannot be further simplified are said to be in their basic form.
Addition and Subtraction of Similar Surds
Surds that are in the basic form can be added and subtracted. Example 2 below will demonstrate the idea.
Simplify √80 +√20 -√45 Solution:
To start with reduce all to their basic forms. That is,√80+√20-√45=√(16X5)+√(4X5)-√(9X5) RHS(the right hand side):=4√ 5 +2√5-3√5 = 6√5 -3√5 = 3√5
Observe that the above surds can be added or subtracted due to the fact that they are in similar form, that is, numbers under the radical signs are equivalent and they possess the same index.
Mixed surds like 2√2 + 2√7 - 2√3 are not similar, so they cannot be added or subtracted, that is, they cannot be simplified further.
Multiplication and Division of Surds
Multiplication and division of surds are done through two basic laws of surds.
For multiplication of surds, the rule is: √a√b=√(a.b)
Examples: 3a)√5 √3 = √(5X3)=√(15) 3b)√ 2√7 = √(2X7)=√(14)
For division of surds the following rule is applicable: √a/√b=√(a/b)
Examples: 4a)'√6/√3=√(6/3)=√2 4b)√18/√3=√(18/3)=√6
Rationalizing the Denominator
A surd like √3/5 cannot be simplified further; but one like 2/√3 can be written in a proper form, because it is not normal to have the radical in the denominator. The process of removing the radical from the denominator is known as rationalization. In order to do rationalization, you are supposed to know about conjugate surds.
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