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In multiplication of surds we will learn how to find the product of two or more surds.
Follow the following steps to find the multiplication of two or more surds.
Step I: Express each surd in its simplest mixed form.
Step II: Observe whether the given surds are of the same order or not.
Step III: If they are of the same order then the required product is obtained by multiplying the product of the rational co-efficient by the product of surd-factors.
If they are of different orders then the product is obtained by the above method after reducing them to surds of the same order.
If different order surds have the same base then their product can easily be obtained using the laws of indices.
Multiplication of surds can be obtained by simply following the law of indices.
aβxΓbβx=x1aΓx1b=x(1a+1b)
From the above equation we can understand that if surds of rational number x are in different orders, then the product of those surds can be obtained by the sum of indices of the surds. Here surds of rational number x are in order a and b, so the indices of the surds are 1a and 1b and after multiplication the result index of x is (1a+1b).
If the surds are in same order, then multiplication of surds can be done by following rule.
aβxΓaβy=aβxy
From the above equation we can understand that if two or more rational numbers like x and y are in a same order a, then product of those surds can be obtained by product of the radicands or rational numbers.
If the surds are not in same order, we can express them in same order to obtain the result of a multiplication problem. But first we should try to express the surds in simplest forms and compare with other surds that they are similar surds or equiradical or dissimilar. Whatever the surds are, we can multiply the rational coefficients. Products of surds can rational or irrational, depending upon the situations.
Like 2β3Γ2β3 = 3, so the product of two similar surds is rational number.
But 2β3Γ3β3 = 3(12+13) = 356
Now we will solve some problems on multiplication of surds to understand more on this.
Examples of multiplication of surds:
1. Find the product of 52β5 and 2β45.
Solution:
52β5 Γ 2β45 = 52β5Γ5Γ3Γ3 = 5 Γ 5 Γ 3 = 75.
2. Find the product of 7β4 and 5β3
Solution:
The product of 7β4 and 5β3
= (7β4) Γ (5β3)
= (7 Γ 5) Γ (β4 Γ β3)
= 35 Γ 4β4β 3
= 35 Γ β12
= 35β12
3. Find the product of 32β2 and 46β3.
Solution:
32β2 and 46β3 are in order 2 and 6. As the LCM of 2 and 6 is 6, we can convert 32β2 into a surd of order 6.
32β2 Γ 46β3 = 3Γ212 Γ 46β3
= 3Γ236 Γ 46β3
= 3Γ816 Γ 46β3
= 36β8 Γ 46β3
= 3 Γ 4 Γ 6β8 Γ 6β3
= 12 Γ 6β8Γ3
= 126β24.
4. Find the product of 2β12, 7β20 and β32
Solution:
The product of 2β12, 7β20 and β32
= (2β12) Γ (7β20) Γ (β32)
= (2β2β 2β 3) Γ (7β2β 2β 5) Γ (β2β 2β 2β 2β 2)
= (4β3) Γ (14β5) Γ (4β2)
= (4 Γ 14 Γ 4) Γ (β3 Γ β5 Γ β2)
= 224 Γ β3β 5β 2
= 224 Γ β30
= 224β30
5. Find the product of 32β12, 2β98 and 52β27.
Solution:
32β12 Γ 2β98 Γ 52β27
= 32β2Γ2Γ3 Γ 2β7Γ7Γ2 Γ 52β3Γ3Γ3
= 122β3 Γ 72β2 Γ 152β3
= 12 Γ 7 Γ 15 Γ 2β3Γ2Γ3
= 1260 Γ 3 Γ 2β2
= 37802β2.
6. Simplify: 2β2 Γ 7β5 Γ 3β3.
Solution:
3β3 Γ 2β2 Γ 7β5
The orders of the given surds are 4, 2, 3 respectively and L.C.M. of 4, 2 and 3 is 12.
β3 = 31/4 = 33/12 = 12β33 = 12β27
β2 = 21/2 = 26/12 = 12β26 = 12β64
β5 = 51/3 = 54/12 = 12β54 = 12β625
Therefore, the given expression 3β3 Γ 2β2 Γ 7β5
= (3 Γ 2 Γ 7) Γ (β3 Γ β2 Γ β5)
= 42 Γ (12β27 Γ 12β64 Γ 12β625)
= 42 Γ (12β27Γ64Γ625)
= 42 Γ (12β1080000)
= 4212β1080000
7. Find the product of 32β2, 53β4 and 24β8.
Solution:
32β2 Γ 53β4 Γ 24β8
= 3 Γ 212 Γ 5 Γ 3β22 Γ 2 Γ 4β23
= 3 Γ 5 Γ 2 Γ 212 Γ 223 Γ 234
= 30 Γ 2(12+23+34)
= 30 Γ 22312
= 3012β223
= 3012β2(12+11)
= 30 Γ 212β211
= 6012β2048.
8. Simplify: 4β3 Γ 2β9 Γ 5β27
Solution:
4β3 Γ 2β9 Γ 5β27
= (4 Γ 2 Γ 5) Γ (31/2 Γ 91/3 Γ 271/4)
= 40 Γ (31/2 Γ 32/3 Γ 33/4)
= 40 Γ 31/2+2/3+3/4
= 40 Γ 323/12
= 40 Γ 12β323
= 40 Γ 12β312β 311
= 40 Γ 312β311
= 12012β177147
9. Find the product of 2βx, 4βx and 2βy.
Solution:
2βx Γ 4βx Γ 2βy
As the surds are in order 2, 4 and 2, their LCM is 4, we need to convert 2βx and 2βy into order 4.
= x24 Γ x14 Γ y24
= x(24+14) Γ 4βy2
= x34 Γ 4βy2
= 4βx3 Γ 4βy2
= 4βx3y2.
11 and 12 Grade Math
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