ADDITION, SUBTRACTION AND MULTIPLICATION OF SURD
SUBJECT: MATHEMATICS
CLASS: SS 3
FIRST TERM
LESSON PLAN WITH SCHEME OF WORK
WEEK 2 DATE………………….
TOPIC: ADDITION, SUBTRACTION AND MULTIPLICATION OF SURD
CONTENTS:
1. Definition of surd
2. Like surds
3. Simplification of surd
4. Addition and subtraction of surd
5. multiplication of surds
DEFINITION OF SURDS
A number which can be expressed as a quotient m/n of two integers, (n≠ 0) is called a rational number. Any real number which is not rational is called IRRATIONAL. Irrational numbers which are in the form of roots are called SURDS.
Like Surds
Another important type of surds are like surds, which are defined as two or more surds that share the same radicand. For example, √2 and √3 are like surds, since both have a radicand of 2. Similarly, √4 and √8 are like surds, since both share a radicand of 2√2.
Simplification of Surds
When simplifying surds, it is important to remember that all operations must be performed on the radicands only. For example, in order to simplify √3 + √5, we would first need to write out the expression as √3/2 + √5/2. Next, we would simplify each of the surds by dividing both sides by 2. The result is (√3 + √5)/2, which is an equivalent form of the original surd.
Examples:
, , π, , are all examples of irrational numbers.
is taken to mean the positive square root of m and n , is called a RADICAL.
LIKE SURD
What are like surds?
Two or more surds are like surds if the number under the square root sign is the same e.g , , , and are like surds.
SIMPLIFICATION OF SURD
Illustration
By putting m = 16 and n = 9, find which of the following pairs of expression are equal.
1. and x
2. and +
3. and
4. and –
5. and
6. and
Solution
1. = = 12 x = 4 x 3 = 12
2. = = 5 + = 4 + 3 = 7
3.
4. = = – = – = 4 – 3 = 1
5. = = 2 x 4 = 8 = = or
6. = = 3 x 3 = 9 = = = 9
Solutions to question 1, 3, and 6 are equal.
Note:- The illustration demonstrates the fact that = x and =
To simplify surd where the square root sign as a product of the factor one of which should be a perfect square, then simplify the surd by taking the square root of the perfect square.
Example1
Simplify
(a) (b) (c)
Solution
(a) = = x =
(b) = = x =
(c) = = 3 x x = 3 x 5 x =
Example 2
Express the following as a single surd
(1) (2)
Solution
1. = = =
2. = = =
Evaluation:
1.Simplify
a. b. c
2. Express the following as a single surd.d. e.
ADDITION AND SUBTRACTION OF SURDS
Two or more surds can be added together or subtracted from one another if they are like surds. Before addition or subtraction, the surds should first be simplified if possible.
Examples
Simplify:-
1. + 2. –
3. + 4. – +
Solution
1. + 2. –
= + = (3 -7)
= + =
= (2+ 1) =
3. – + 0
= – +
= – +
= – +
= (4-12 + 20)
= 12
4. – +
= – +
= – + 2 x 5
= – +
= (- 3+10)
= +
7 = 6.
= 7 + 2 x 5 = 20
SOLUTION OF RADICAL EQUATION
A radical equation is an equation containing a square root sign. If the square root sign is insignificant, the equation can be solved by factorization method. But if the sqaure root sign is significant and cannot be pulled out, the equation is called a radical equation. Usually if the square root sign has a single term on the right side of the equation, it can be solved by either factorization method or application of quadratic formula. But in case the equation involves more than one term under square root sign and cannot be solved by factorization method alone, then application of quadratic formula is required.
Example 1
Solve for x. x + 3 = 2 + 3
Solution
Let the surd be made rational by multiplying both sides by 6 and applying the factorization method, we get -2 + 9 = 18 or 0 = 20
Applying the quadratic formula to solve for x, we get -2 + 9 = 18 or 0 = 20. Thus x = -10 and 8 are two value of x that satisfy the equation.
Example 2
Solve for x. 4x – 3 = 5 + 8
Solution
Let the surd be made rational by multiplying both sides by 2, we get -8 + 3 = 10 or = 18
Applying the quadratic formula to solve for x, we get -2 + 3 = 5 or 0 = 18. Thus x = -13 and 13 are two values of x that satisfy the equation.
Solve for x. 2x – 1 = 3 + 6
Solution
Let the surd be made rational by multiplying both sides by 3, we get 6 – 3 = 9 or 0 = 18
Applying the quadratic formula to solve for x, we get b = 5 and c = 12. Thus if a = 3, then the value of x = 4. Thus x = 4 is one value of x that satisfies the equation.
Evaluation:
1. Use factorization method to solve for x if possible or use quadratic formula (ax2 + bx + c) = 0
(a) 2x – 1 = 3 + 6
(b) 4x – 3 = 5 + 8
(c) 2x – 1 = 3 + 6
2. Illustrate the steps in solving a radical equation using quadratic formula
a. 2x – 1 = 3 + 6
Solve for x by applying the quadratic formula
ax2 + bx + c = 0, a = 2, b = -1 and c = 6.
We get two values for x: 5 and 4 since D = [-b ± √(b2 – 4ac)]/2a = (-(-1) ± √((-1
Evaluation:
Simplify the following
1. – – 2. – 3. 6 + +
4. Simplify the following + –
5. Express as a single surd (i) 7√2 (ii) 8√7
MULTIPLICATION OF SURDS
When two or more surds are multiplied together, they should first be simplified if possible, then multiply whole number with whole number and surd with surd.
Examples:
Simplify (1) x (2) )2(3) x x
Solution
(1) x 2. ( )2
= x = x
= x = (4 x 4)
= 3 x 5 x = 16 x
= 15 x = 16 x 3 = 48
=
(3) x x
= x x
= x x
= 4 x
= 4 x
= 4 x 2
=
Evaluation:
Simplify:
1. x x 2. x x 3. )3 4. (2√5)3 5. (3√7 – √3)(3√7 + √3)
FURTHER EXAMPLES ON MULTIPLICATION OF SURDS
(1) (2)
Solution
(1)
=
=
=
=
(2)
=
=
=
=
Evaluation: Simplify: 1. (5√6 – 3)(5√6 + 3) 2. (7√13 + √5)(7√13 + √5)
DIVISION OF SURDS (RATIONALIZATION)
1. Division of Surd (Rationalization)
2. Multiplication of Surd involving and conjugate of binomial surd.
3. Expression with Surds in Both Denominator and Numerator.
4. Expression involving surds in Exponent
5. Expressed as a single Surd (Reducing to Lowest Terms).
Example 1:
Simplify the expression: 5√6 – 3/2√7 + 4√5
Solution:
The expression can be simplified by dividing all the surds in the denominator by their conjugate.
Thus, 5√6 – 3/2√7 + 4√5 = (5√ 6)/(2√ 7) – (3/2√ 7)/(2√ 7) + (4√ 5)/(2√ 7).
Hence, the given expression is equal to 2√ 3 – 3/2 + 8/7.
Example 2: Rationalize the denominator.
Solution: The following steps should be followed in rationalizing a surd in the denominator.
(i) Multiply the denominator and numerator by a conjugate of the denominator to convert all surds into rationals;
(ii) Simplify the resulting expression in step (i).
Hence, (√13)(√9/2) = 3√13 + 2√9 = 3√13/2 + 2√9/2 = (3 + 2)√13/2 = 5√13/2.
Hence, the given expression is equal to 5√6 / 2.
Example 3: Simplify the following expression and express it as a single surd
2√7 + 7√5 – 5√12
Solution
The expression can be rationalized by multiplying the denominator and numerator by its conjugate. Thus, 2√7 + (√35 – 10√6)/(√ 35 – 10 √6) + (√35 – 5√12)/(√ 35 – 5 √12)
= (2 + 7√5 – 5√12)/(35 – 12) = (9 – 5√12)/23.
Hence, the given expression is equal to the surd 9/23.
Examples
1. = 3 2.
Note: When a fraction has Surd as the denomination, the Surd is rationalized.
To rationalize the denominator of a rational surd, means to rationalize denominator remove the square root sign from the denominator.
Two types are considered
(a) Monomial denominator
(i) (ii)
(iii)
(iv)
EVALUATION
(1) (2) (3)
BINOMIAL DENOMINATOR:
Conjugate of a binominal surd
When an expression contains two terms e.g. and one or both of them contains surd, it is known as BINOMINAL SURD.
In such expressions, each term must be multiplied by a conjugate of its denominator to rationalize it.
The product of the two terms is called CONJUGATE OF BINOMINAL SURD and is denoted by ‘ ‘.
(2) (3) .(4) +(5) +
Note: In simplification, each surd should be simplified separately before they are rationalized.
Example 1: Reduce the following expression to lowest terms: (√13/25)(15/√10) + √(27/2)
Solution: The given expression can be simplified by multiplying the denominator and numerator by its conjugate. Thus, (√13/25)(15/√10) + √(27/2) = (√13 x 25)/(25x 10) + √(2 x 27)/(2 x 25)
= (√13×25+2×27)/(25×10+50) = (√195 + 2 x 54)/75 = (√245/75).
Hence, the given expression is equal to 3/5.
To rationalize binominal surd, multiply it by its conjugate i.e change the sign of the denominator and use the result to multiply both the numerator and the denominator.
e.g. is the Conjugate of i.e. the sign between the terms is changed or reversed.
Example:
Simplify
(1) (2) (3) 2
Solution
(1)
=
=
(2)
=
=
=
=
=
(3) 2
=
=
=
Evaluation: Rationalize the following 1. 2. 3.
EVALUATION OF EXPRESSION WITH SURDS:
When evaluating a fraction containing a surd, it is advisable to rationalize its denominator.
Example:
Given that and
Evaluate:
(1) (2)
Solution
(1) (2)
,
3. Without using tables or calculator, evaluate:
Solution
=
=
=
= =
Evaluation
- Given that
- Evaluate: (1) (2)
General Evaluation And Revision Questions
1. Evaluate √20 × (√5)3
2.√1.225 = 1.107, √12.25 =3.5 and √100 = 10. Evaluate √1225
3.Given that √2 = 1.414 , √3 =1.732 and √5 =2.236,without using tables or a calculator, evaluate the
following to 2 d.p a. (√18 + √6 – 3) √2 b. √3 (√9 + 3 √27)
4.Calculate the altitude of of an equilateral triangle of side 12cm.Leave your answer in surd form.
5.The angle of depression of a boat from the top of a cliff 11 m high is 300.How far is the boat from the foot of the cliff ? Leave your answer in surd form.
6. Evaluate the following:
Simplify:
1. – + + 2. x x 3.( )5 4. 5. 8√2 – 3√6 + √50
Reading Assignment:
1. Essential Mathematics for SS 2 – by A. J. S. Oluwasanmi pages 25-35.
2. Exam focus (Mathematics) By Ilori & Co
Weekend Assignment
1. Evaluate: (a) 8 (b) (c) (d)
2. …………….is an example of Binominal Surd. (a) (b) (c) (d)
3. Simplify: (a) 3 (b) 3 (c) 6 (d) 12
4. Rationalize (a) (b) (c) (d)
5. If , find Cos x – Sin x such that 0O = x ≤ 90O
(a) (b) (c) (d)
6. Find the Value of: √5 / 3 (a) (b) (c) – 12/9
7. If a=2 and b=√3+1,find (a+b)2 (a) (b) 8.5‑3.5
8. Simplify: x / √3x + 1 (a) 2––1 (b) 1/√3+1
THEORY
1 Express 3√5 – √3 in the form a√15 +b, where a and b are rational numbers.
2√5 +3√3
2. Evaluate the following
(a) (b)
.
