Loading [MathJax]/jax/output/HTML-CSS/fonts/TeX/fontdata.js

Subscribe to our YouTube channel for the latest videos, updates, and tips.


Multiplication of Surds

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.

ax×bx=x1a×x1b=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.

ax×ay=axy

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 23×23 = 3, so the product of two similar surds is rational number.

But 23×33 = 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 525 and 245.

Solution:

525 × 245 = 525×5×3×3 = 5 × 5 × 3 = 75.

2. Find the product of 7∜4 and 5∜3

Solution:

 The product of 7∜4 and 5∜3

= (7∜4) × (5∜3)

= (7 × 5) × (∜4 × ∜3)

= 35 × 443

= 35 × ∜12

= 35∜12


3. Find the product of 322 and 463.

Solution:

322 and 463 are in order 2 and 6. As the LCM of 2 and 6 is 6, we can convert 322 into a surd of order 6.

322 × 463 = 3×212 × 463

= 3×236 × 463

= 3×816 × 463

= 368 × 463

= 3 × 4 × 68 × 63

= 12 × 68×3

= 12624.


4. Find the product of 2√12, 7√20 and √32

Solution:

The product of 2√12, 7√20 and √32

= (2√12) × (7√20) × (√32)

= (2223) × (7225) × (22222)

= (4√3) × (14√5) × (4√2)

= (4 × 14 × 4) × (√3 × √5 × √2)

= 224 × 352

= 224 × √30

= 224√30


5. Find the product of 3212, 298 and 5227.

Solution:

3212 × 298 × 5227

= 322×2×3 × 27×7×2 × 523×3×3

= 1223 × 722 × 1523

= 12 × 7 × 15 × 23×2×3

= 1260 × 3 × 22

= 378022.


6. Simplify: 2√2 × 7∛5 × 3∜3.

Solution:

3∜3 × 2√2 × 7∛5

The orders of the given surds are 4, 2, 3 respectively and L.C.M. of 4, 2 and 3 is 12.

∜3 = 31/4 = 33/12 = 1233 = 1227

√2 = 21/2 = 26/12 = 1226 = 1264

∛5 = 51/3 = 54/12 = 1254 = 12625

Therefore, the given expression 3∜3 × 2√2 × 7∛5

= (3 × 2 × 7) × (∜3 × √2 × ∛5)

= 42 × (1227 × 1264 × 12625)

= 42 × (1227×64×625)

= 42 × (121080000)

= 42121080000

 

7. Find the product of 322, 534 and 248.

Solution:

322 × 534 × 248

= 3 × 212 × 5 × 322 × 2 × 423

= 3 × 5 × 2 × 212 × 223 × 234

= 30 × 2(12+23+34)

= 30 × 22312

= 3012223

= 30122(12+11)

= 30 × 212211

= 60122048.


8. Simplify: 4√3 × 2∛9 × 5∜27

Solution:

4√3 × 2∛9 × 5∜27

= (4 × 2 × 5) × (31/2 × 91/3 × 271/4)

= 40 × (31/2 × 32/3 × 33/4)

= 40 × 31/2+2/3+3/4

= 40 × 323/12

= 40 × 12323

= 40 × 12312311

= 40 × 312311

= 12012177147


9. Find the product of 2x, 4x and 2y.

Solution:

2x × 4x × 2y

As the surds are in order 2, 4 and 2, their LCM is 4, we need to convert 2x and 2y into order 4.

= x24 × x14 × y24

= x(24+14) × 4y2

= x34 × 4y2

= 4x3 × 4y2

= 4x3y2.






11 and 12 Grade Math

From Multiplication of Surds to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Worksheet on Average | Word Problem on Average | Questions on Average

    May 19, 25 02:53 PM

    Worksheet on Average
    In worksheet on average we will solve different types of questions on the concept of average, calculating the average of the given quantities and application of average in different problems.

    Read More

  2. 8 Times Table | Multiplication Table of 8 | Read Eight Times Table

    May 18, 25 04:33 PM

    Printable eight times table
    In 8 times table we will memorize the multiplication table. Printable multiplication table is also available for the homeschoolers. 8 × 0 = 0 8 × 1 = 8 8 × 2 = 16 8 × 3 = 24 8 × 4 = 32 8 × 5 = 40

    Read More

  3. How to Find the Average in Math? | What Does Average Mean? |Definition

    May 17, 25 04:04 PM

    Average 2
    Average means a number which is between the largest and the smallest number. Average can be calculated only for similar quantities and not for dissimilar quantities.

    Read More

  4. Problems Based on Average | Word Problems |Calculating Arithmetic Mean

    May 17, 25 03:47 PM

    Here we will learn to solve the three important types of word problems based on average. The questions are mainly based on average or mean, weighted average and average speed.

    Read More

  5. Rounding Decimals | How to Round a Decimal? | Rounding off Decimal

    May 16, 25 11:13 AM

    Round off to Nearest One
    Rounding decimals are frequently used in our daily life mainly for calculating the cost of the items. In mathematics rounding off decimal is a technique used to estimate or to find the approximate

    Read More