Introduction to surd online
Expressions like `sqrt(16)` ,`root(3)(27)` ,etc. have exact numerical value.They are terminating and rational numbers.These numbers are perfect roots (square root and cube root). But expressions such as `sqrt(2)`,`root(3)(3)` , etc.cannot be written as exact numeric value. Such numbers are called irrational and it is convenient to leave them in the form as in Decimal form they would go on non-terminating. These are called surds. Surds are numbers of the form `root(n)(k)` ,where k is not a perfect nth power of any number.`root(n)(k)` is a surd of nth order.
Ex 1: `sqrt(2)` is a surd of 2nd order.
`root(3)(3)`is a surd of 3rd order.
`root(4)(5)`is a surd of 4th order
A surd has infinite number of non-terminating decimals.Surds are always irrational.
Ex 2: Which of the following numbers are surds?
`sqrt(7)`
`sqrt(9)`
`root(3)(27)`
`sqrt(5)`
`sqrt(100)`
`root(4)(16)`
Solution :
`sqrt(7)` is a surd.
`sqrt(9)` is not a surd because `sqrt(9)` = 3
`root(3)(27)` is not a surd because `root(3)(27)` = 3
`sqrt(5)` is a surd.
`sqrt(100)` is not a surd because `sqrt(100)` = 10
`root(4)(16)` is not a surd because `root(4)(16)` = 2
Simplifying a surd:
Consider the entire surd `sqrt(18)` , it has 9 as one of its factors which is a perfect square.So `sqrt(18)` can be expressed as a product of rational number and a surd.
`sqrt(18)` = `sqrt( 9 * 2)`
= `sqrt(9)` *`sqrt(2)`
= 3 *`sqrt(2)`
= 3`sqrt(2)` This is called a mixed surd.
Ex 3: Express the mixed surd 4`sqrt(2)` as an entire surd.
Solution: 4`sqrt(2)` =`sqrt(16)` * `sqrt(2)` because 4 = `sqrt(16)`
= `sqrt( 16 * 2)`
= `sqrt(32)`
Addition , Subtraction and Multiplication on surd online
Addition and subtraction of surds is simple, however operation can be performed only on surds of same order and the radical must be same.That is number inside the root must be the same.Lets consider an example:
Ex 1: 5`sqrt(11)` + 2`sqrt(11)` = `sqrt(11)` ( 5 + 2) = 7`sqrt(11)`
Ex 2: 6`sqrt(5)` - 3`sqrt(5)` = `sqrt(5)` ( 6 - 3) = 3`sqrt(5)`
Ex 3: Simplify 6`sqrt(3)` + `sqrt(75)`
Solution : First simplify `sqrt(75)` .Express it as mixed surd.
`sqrt(75)` = `sqrt(25 * 3)` = 5`sqrt(3)`
Then, 6`sqrt(3)` + 5`sqrt(3)` = 11`sqrt(3)`
Multiplication of similar surds gives rational number. i.e. `sqrt(x)` * `sqrt(x)` = x
Ex 4: Simplify (`sqrt(6)`)2
Solution: (`sqrt(6)` )2 = `sqrt(6)` * `sqrt(6)`
= `sqrt(6 * 6)`
= 6
Multiplication of unlike surds give irrational number.
Ex 5: Simplify `sqrt(8)` * `sqrt(3)`
Solution: `sqrt(8)` * `sqrt(3)` = `sqrt(8 * 3)`
= `sqrt(24)`
= `sqrt(4 * 6)`
= 2`sqrt(6)`
Ex 6: Simplify 3`sqrt(5)` * 5`sqrt(3)`
Solution:
Method 1: 3`sqrt(5)` * 5`sqrt(3)` = `sqrt(3 * 3 * 5)` * `sqrt(5 * 5 * 3)`
= `sqrt(3 * 3 * 5 * 5 * 5 * 3)`
= 3 * 5 `sqrt(15)`
= 15`sqrt(15)`
Method 2: 3`sqrt(5)` * 5`sqrt(3)` = 3 * 5 *`sqrt(5)` * `sqrt(3)`
= 15`sqrt(5 * 3)`
= 15`sqrt(15)`
Ex 7: Simplify 2`sqrt(18)` * 3`sqrt(20)`
Solution: 2`sqrt(18)` * 3`sqrt(20)` = 2`sqrt(9 * 2)` * 3`sqrt(4 * 5)`
= 2*3`sqrt(2)` * 3*2`sqrt(5)`
= 6*6*`sqrt(2)` * `sqrt(5)`
= 36`sqrt(2*5)`
= 36`sqrt(10)`
Rationalisation of the denomiator on surd online
We express a radical fraction, such as `sqrt(5)` /`sqrt(2)` in a form that has rational denominator.
Ex 1: Simplify `sqrt(5)` / `sqrt(2)`
Solution: In order to rationalise we multiply numerator and denominator by `sqrt(2)`
`(sqrt(5))/(sqrt(2))` * `(sqrt(2))/(sqrt(2))`= `(sqrt(5 * 2))/(2)` = `(sqrt(10))/(2)`
Ex 2: simplify `(sqrt(35))/(sqrt(15))`
Solution: `(sqrt(35))/(sqrt(15))` * `(sqrt(15))/(sqrt(15))` = `(sqrt(7 * 5 * 5 * 3))/(sqrt(15 * 15))` = `(5sqrt(21))/(15)` = `(sqrt(21))/(3)`
Consider (2 + `sqrt(3)`) * ( 2 - `sqrt(3)`) = 4 - 3 = 1. Their product is a rational number. Hence 2 - `sqrt(3)` is called the conjugate of the surd 2+`sqrt(3)` or vice verse.`sqrt(a)` + `sqrt(b)` is called the conjugate of `sqrt(a)` - `sqrt(b)` if their product is a rational number
Ex 3 : Simplify `(1)/(sqrt(2) + sqrt(3))`
Solution: Here we have to rationalise the denominator.Hence multiply and divide numerator and denominator by `sqrt(2)`-`sqrt(3)`(the opposite sign of the denominator)
We have, `(1)/(sqrt(2) + sqrt(3))` * `(sqrt(2) - sqrt(3))/(sqrt(2) - sqrt(3))` = `(sqrt(2)- sqrt(3))/(2-3)` = `sqrt(3)` - `sqrt(2)`
Expressions like `sqrt(16)` ,`root(3)(27)` ,etc. have exact numerical value.They are terminating and rational numbers.These numbers are perfect roots (square root and cube root). But expressions such as `sqrt(2)`,`root(3)(3)` , etc.cannot be written as exact numeric value. Such numbers are called irrational and it is convenient to leave them in the form as in Decimal form they would go on non-terminating. These are called surds. Surds are numbers of the form `root(n)(k)` ,where k is not a perfect nth power of any number.`root(n)(k)` is a surd of nth order.
Ex 1: `sqrt(2)` is a surd of 2nd order.
`root(3)(3)`is a surd of 3rd order.
`root(4)(5)`is a surd of 4th order
A surd has infinite number of non-terminating decimals.Surds are always irrational.
Ex 2: Which of the following numbers are surds?
`sqrt(7)`
`sqrt(9)`
`root(3)(27)`
`sqrt(5)`
`sqrt(100)`
`root(4)(16)`
Solution :
`sqrt(7)` is a surd.
`sqrt(9)` is not a surd because `sqrt(9)` = 3
`root(3)(27)` is not a surd because `root(3)(27)` = 3
`sqrt(5)` is a surd.
`sqrt(100)` is not a surd because `sqrt(100)` = 10
`root(4)(16)` is not a surd because `root(4)(16)` = 2
Simplifying a surd:
Consider the entire surd `sqrt(18)` , it has 9 as one of its factors which is a perfect square.So `sqrt(18)` can be expressed as a product of rational number and a surd.
`sqrt(18)` = `sqrt( 9 * 2)`
= `sqrt(9)` *`sqrt(2)`
= 3 *`sqrt(2)`
= 3`sqrt(2)` This is called a mixed surd.
Ex 3: Express the mixed surd 4`sqrt(2)` as an entire surd.
Solution: 4`sqrt(2)` =`sqrt(16)` * `sqrt(2)` because 4 = `sqrt(16)`
= `sqrt( 16 * 2)`
= `sqrt(32)`
Addition , Subtraction and Multiplication on surd online
Addition and subtraction of surds is simple, however operation can be performed only on surds of same order and the radical must be same.That is number inside the root must be the same.Lets consider an example:
Ex 1: 5`sqrt(11)` + 2`sqrt(11)` = `sqrt(11)` ( 5 + 2) = 7`sqrt(11)`
Ex 2: 6`sqrt(5)` - 3`sqrt(5)` = `sqrt(5)` ( 6 - 3) = 3`sqrt(5)`
Ex 3: Simplify 6`sqrt(3)` + `sqrt(75)`
Solution : First simplify `sqrt(75)` .Express it as mixed surd.
`sqrt(75)` = `sqrt(25 * 3)` = 5`sqrt(3)`
Then, 6`sqrt(3)` + 5`sqrt(3)` = 11`sqrt(3)`
Multiplication of similar surds gives rational number. i.e. `sqrt(x)` * `sqrt(x)` = x
Ex 4: Simplify (`sqrt(6)`)2
Solution: (`sqrt(6)` )2 = `sqrt(6)` * `sqrt(6)`
= `sqrt(6 * 6)`
= 6
Multiplication of unlike surds give irrational number.
Ex 5: Simplify `sqrt(8)` * `sqrt(3)`
Solution: `sqrt(8)` * `sqrt(3)` = `sqrt(8 * 3)`
= `sqrt(24)`
= `sqrt(4 * 6)`
= 2`sqrt(6)`
Ex 6: Simplify 3`sqrt(5)` * 5`sqrt(3)`
Solution:
Method 1: 3`sqrt(5)` * 5`sqrt(3)` = `sqrt(3 * 3 * 5)` * `sqrt(5 * 5 * 3)`
= `sqrt(3 * 3 * 5 * 5 * 5 * 3)`
= 3 * 5 `sqrt(15)`
= 15`sqrt(15)`
Method 2: 3`sqrt(5)` * 5`sqrt(3)` = 3 * 5 *`sqrt(5)` * `sqrt(3)`
= 15`sqrt(5 * 3)`
= 15`sqrt(15)`
Ex 7: Simplify 2`sqrt(18)` * 3`sqrt(20)`
Solution: 2`sqrt(18)` * 3`sqrt(20)` = 2`sqrt(9 * 2)` * 3`sqrt(4 * 5)`
= 2*3`sqrt(2)` * 3*2`sqrt(5)`
= 6*6*`sqrt(2)` * `sqrt(5)`
= 36`sqrt(2*5)`
= 36`sqrt(10)`
Rationalisation of the denomiator on surd online
We express a radical fraction, such as `sqrt(5)` /`sqrt(2)` in a form that has rational denominator.
Ex 1: Simplify `sqrt(5)` / `sqrt(2)`
Solution: In order to rationalise we multiply numerator and denominator by `sqrt(2)`
`(sqrt(5))/(sqrt(2))` * `(sqrt(2))/(sqrt(2))`= `(sqrt(5 * 2))/(2)` = `(sqrt(10))/(2)`
Ex 2: simplify `(sqrt(35))/(sqrt(15))`
Solution: `(sqrt(35))/(sqrt(15))` * `(sqrt(15))/(sqrt(15))` = `(sqrt(7 * 5 * 5 * 3))/(sqrt(15 * 15))` = `(5sqrt(21))/(15)` = `(sqrt(21))/(3)`
Consider (2 + `sqrt(3)`) * ( 2 - `sqrt(3)`) = 4 - 3 = 1. Their product is a rational number. Hence 2 - `sqrt(3)` is called the conjugate of the surd 2+`sqrt(3)` or vice verse.`sqrt(a)` + `sqrt(b)` is called the conjugate of `sqrt(a)` - `sqrt(b)` if their product is a rational number
Ex 3 : Simplify `(1)/(sqrt(2) + sqrt(3))`
Solution: Here we have to rationalise the denominator.Hence multiply and divide numerator and denominator by `sqrt(2)`-`sqrt(3)`(the opposite sign of the denominator)
We have, `(1)/(sqrt(2) + sqrt(3))` * `(sqrt(2) - sqrt(3))/(sqrt(2) - sqrt(3))` = `(sqrt(2)- sqrt(3))/(2-3)` = `sqrt(3)` - `sqrt(2)`
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