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The numbers whose exact roots cannot be found are called surds. Such are the special type of irrational number under radical sign like √5, √21, √33, etc. The sign √ is called radical sign. In \(\sqrt[n]{a}\), n is the degree of the surd and a is called radicand. \(\sqrt[n]{a}\) can be written as a^{\(\frac{1}{n}\)} and\(\sqrt[n]{a}\) is called the n^{th} root a.
Similarly, √3 is a surd of order 2, \(\sqrt[3]{16}\) is a surd of order 3 and \(\sqrt[n]{a}\) is a surd of order 'n'. The surds which have no relation are called pure surds. √2,√3,√5 etc. are some examples of pure surds. Surds having a rational factor and a surds factor are called mixed surds 3√2, 5√5, etc.are some examples of mixed surds.
When a surd consists of only one term, it is called a simple surd. A surd having two or more terms is called compound surd. In other words, the algebraic sum of two or more surd is called a compound surd. √2+√5, √5+ 3, 4-√7 are some examples of compound surd.
The algebraic sum of three surds like √2+√3+√5 or the algebraic sum of two surds and a rational quantity like √2+√3-5 is known as a trinomial surd. Two surds having the same radicand and same order are called like surd. 2√3, √3, 7√3 are like surds. The surds which are not like i.e. the order or the radicands of surds are differentare called, unlike surds. √5,3√7,5√3 are unlike surds. Two binomial surds different only in sign connecting their terms are called conjugate to each other. So, √3+√5 and √3-√5 are conjugate to each other. Surds of order 3 are called a quadratic surd and a surd of order 3 is called a cubic surd. √5,√11,√7 are quadratic surds and \(\sqrt[3]{5}\) , \(\sqrt[3]{7}\), \(\sqrt[3]{11}\) cubic surds.
Addition and subtraction
Only the like surds can be added or subtracted unlike surds and neither be added nor be subtracted. To add or subtract like surds, they should be expressed in their simplify: 3√20 + √45+√80
Solutions:
3√20 + √45+√80
= 3\(\sqrt{4×5}\) +\(\sqrt{9×5}\) +\(\sqrt{16×5}\)
=3× 2√5 + 3√5 + 4\(\sqrt{5}\)
=6√5 + 3√5 + 4√5
=13√5
Multiplication and division of Surds
Multiplication and division of surds are possible when they are of same order. If the surds are of the different order, we have to change them so that their order become same. Before multiplying, surds are expressed in their simplest form, while multiplying ,surds are multiplied with surds and rational are multiplied with rational.
If the product of two surds is rational, then each of them is called the rationalizing factor of the other. The product of √3+√2 and √3-√2 is 3 which is a rationalizing factor of √3-√2 and vice-versa. √3+√2 and√3-√2 are conjugate to each other. So, for a binomial quadratic surd, its conjugate is the rationalizing factor. Note that √5 is rationalized when it is multiplied by √5 so √5 is rationalizingfactor of √5.
Since 2√5 x√5 = 10, then
2√5 and √5 are rationalizing factor of each other.
Let us took at another example.
2√5 x 3√5 =30
∴ 2√5 and 3√5 are rational factors of each other. Thus, surd may have more than one rational factor. But it is better to choose a rational factor that is easy to evaluate.
The equations are which the variables are expressed in terms of surds are called the equations involving surds.\(\sqrt{3x+2}\) +\(\sqrt{3x-11}\) =9 is an example of equation involving surds.
The steps of solving surd equations are as follows:
In the solution of equation involving radical sign, verification must be shown. If any value of the variable doesn't satisfy the given equation, it should be discarded.
Soln:
If CA=1 unit and CB=2 units of a right angledΔACB,then
AB^{2}=CA^{2}+CB^{2}=1^{2}+2^{2}=5
∴ AB=\(\sqrt{5}\)
So, an arc of radius equal to AB cuts the number line at F so that AF=\(\sqrt{5}\). Similarly, in right angledΔACD, take AD=\(\sqrt{3}\) units are the radius which cuts the number line at E. So that, AE=\(\sqrt{3}\). The graph \(\sqrt{5}\) and \(\sqrt{3}\) are show on the number line alongside.
Here,
\(\frac{10}{3}\),\(\frac{13}{4}\),\(\frac{16}{5}\) and\(\frac{19}{6}\)etc are rational numbers between 3 and 4 because division of 10 by 3,13 by 4, 16 by 5 and 19 by 6 give the repeating decimal or terminating decimal numbers between 3 and 4.
Here,
\(\sqrt{11}\)
=\(\sqrt[2]{11}\) ∴Degree of\(\sqrt{11}\)=2.Ans.
Here,
Degree of\(\sqrt[3]{5}\)=3.Ans.
Here,
5\(\sqrt{3}\)=\(\sqrt{25}\).\(\sqrt{3}\)
=\(\sqrt{25×3}\)
=\(\sqrt{75}\).Ans.
Here,
\(\sqrt{90}\)=\(\sqrt{9×10}\)
=3\(\sqrt{10}\).Ans.
Here,
\(\sqrt[3]{375}\)
=\(\sqrt[3]{125×3}\)
=\(\sqrt[3]{5^3×3}\)
=5\(\sqrt[3]{3}\).Ans.
Here,
\(\sqrt[4]{5}\)=\(\sqrt[4×3]{5^3}\) =\(\sqrt[12]{125}\)
Again,\(\sqrt[3]{4}\)=\(\sqrt[3×4]{4^4}\)=\(\sqrt[12]{256}\)
Here,125‹ 256
or,\(\sqrt[12]{125}\)‹,\(\sqrt[12]{256}\)
∴\(\sqrt[4]{5}\) ‹ \(\sqrt[3]{4}\).Ans.
Here,
\(\sqrt[5]{4×4×4}\)
=\(\sqrt[5]{2×2×2×2×2×2}\)
=\(\sqrt[5]{2^5×2}\)
=2\(\sqrt[5]{2}\).Ans.
Here,\(\sqrt[3]{3}\)×\(\sqrt[3]{4}\)
=\(\sqrt[3]{12}\).Ans.
Here,
\(\sqrt[3]{7}\)+\(\sqrt[12]{7}\)
=\(\sqrt[15]{7}\).Ans.
Here,
=8\(\sqrt[3]{5}\)-3\(\sqrt[3]{5}\)
=5\(\sqrt[3]{5}\).Ans.
Here, Multiplying2\(\sqrt{3}\) by \(\sqrt{3}\) we get,
2\(\sqrt{3}\)×\(\sqrt{3}\)=2×3=6 which is rational number
∴ Rational factor of2\(\sqrt{3}\)=\(\sqrt{3}\.Ans.
Simplify
√27 + √75 - 8√3
Ten
Zero
0ne
Two
Simplify
√32 + √8 - √72
Zero
Two
Three
Four
Simplify
√12 -√75 +√48
√4
√9
√3
√1
Simplify
√50 + √18 - 8√2
Ten
Five
Four
Zero
Simplify
√125 - √45 + √5
3√9
3√5
3√6
3√1
Simplify
3√27 + 2√12 - 2√3
21√3
11√2
11√3
12√3
Simplify
4√45 - 3√20 + 8√5
14√4
10√5
11√5
14√5
Simplify
3√2+ 4√2500+ 4√64 + 6√8
22√2
22√8
20√2
22√6
Solve
(sqrt {x-7}) =√x-1
16
8
9
12
Solve
√x -1 = (sqrt {x-5})
9
3
15
10
Solve
(sqrt {x-7})=7- √x
12
15
16
8
Solve
√x+(sqrt {x-20})=10
12
36
29
30
Solve
(sqrt {x+24})- √x=2
22
12
19
25
Solve
2√x -(sqrt {4x-11})=1
9
1
6
16
Solve
(sqrt{4}{2x+3})=3
23
39
35
44
ASK ANY QUESTION ON Surds
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