Division of Surds

In division of surds we need to divide a given surd by another surd the quotient is first expressed as a fraction. Then by rationalizing the denominator the required quotient is obtained with a rational denominator. For this the numerator and the denominator are multiplied by appropriate rationalizing factor. In rationalization of surds the multiplying surd-factor is called the rationalizing factor of the given surd.

Division of surds in general can be obtained by following the law of indices.

\(\sqrt[a]{x}\div \sqrt[b]{x}\)= \(\frac{\sqrt[a]{x}}{\sqrt[b]{x}}\)= \(x^{(\frac{1}{a}-\frac{1}{b})}\).

From the above equation we can understand that if surds of rational number x are in different orders, then the indices are expressed in fraction and division is obtained by the subtraction of indices of the surds. Here surds of rational number x are in order a and b, so the indices of the surds are \(\frac{1}{a}\) and \(\frac{1}{b}\) and after division the result index of x is \({(\frac{1}{a}-\frac{1}{b})}\).

If the surds are in same order, then division of surds can be done by following rule.

\(\sqrt[a]{x}\div \sqrt[a]{y}\)= \(\frac{\sqrt[a]{x}}{\sqrt[a]{y}}\)= \(\sqrt[a]{\frac{x}{y}}\).

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 division of those surds can be obtained by division of the radicands or rational numbers of the surds.

In division if the surds are not in same order, we can convert them in same order to obtain the result of a division 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.

Sometimes for division of surds, we need to rationalize the denominator to get a simpler form and obtain a result. For this both numerator and denominator need to be multiplied by appropriate rationalizing factor.

Like for example \(\frac{\sqrt[2]{x}}{\sqrt[2]{y}}\)

= \(\frac{\sqrt[2]{x}\times \sqrt[2]{y}}{\sqrt[2]{y}\times \sqrt[2]{y}}\) 

=\(\frac{\sqrt[2]{xy}}{y}\)

In the above example \(\sqrt[2]{y}\) is the denominator and rationalizing factor for \(\sqrt[2]{y}\) is \(\sqrt[2]{y}\). So \(\sqrt[2]{y}\) is multiplied to both the nominator and denominator to rationalize the surd.


Now we will solve some problems to understand more on division of surds:

1. Find the division of \(\sqrt[2]{12}\) by \(\sqrt[2]{3}\).

Solution:

\(\sqrt[2]{12}\) ÷ \(\sqrt[2]{3}\)

= \(\sqrt[2]{\frac{12}{3}}\)

= \(\sqrt[2]{\frac{4\times 3}{3}}\)

= \(\sqrt[2]{4}\)

= \(\sqrt[2]{2^{2}}\)

= 2.   


2. Divide: √x by √y

Solution:

√x by √y

= √x ÷ √y

= √x/√y

= \(\sqrt{\frac{x}{y}}\)


3. Find the division of \(\sqrt[2]{5}\) by \(\sqrt[2]{3}\).

Solution:

\(\sqrt[2]{5}\) ÷ \(\sqrt[2]{3}\)

= \(\frac{\sqrt[2]{5}}{\sqrt[2]{3}}\)

= \(\frac{\sqrt[2]{5}\times \sqrt[2]{3}}{\sqrt[2]{3}\times \sqrt[2]{3}}\) ….multiplying \(\sqrt[2]{3}\) as rationalizing factor

= \(\frac{\sqrt[2]{15}}{3}\).


4. Divide the first surd by the second surd: √32, √8

Solution:

√32 divided by √8

= √32 ÷ √8

= \(\sqrt{\frac{32}{8}}\)

= √4

= 2.


5. Find the division of \(\sqrt[2]{3}\) by \(\sqrt[2]{2}-1\).

Solution:

\(\sqrt[2]{3}\) ÷ \(\sqrt[2]{2} - 1\)

= \(\frac{\sqrt[2]{3}}{\sqrt[2]{2} - 1}\)

As the denominator is \(\sqrt[2]{2} - 1\), for the division, we need to multiply it with a rationalizing factor \(\sqrt[2]{2} + 1\).

= \(\frac{\sqrt[2]{3}(\sqrt[2]{2} + 1)}{(\sqrt[2]{2} - 1)(\sqrt[2]{2} + 1)}\)

= \(\frac{\sqrt[2]{3}\times \sqrt[2]{2} + \sqrt[2]{3}}{2 - 1}\)….. as we know \((a + b)(a - b) = a^{2} - b^{2}\)

=  \(\sqrt[2]{6}\) + \(\sqrt[2]{3}\).


6. Find the quotient dividing the surd √96 by the surd √16.

Solution:

Required quotient

= √96 ÷ √16

= \(\sqrt{\frac{96}{16}}\)

= √6.


7. Find the division of \((x-1)\) by \(\sqrt[2]{x}-1\).

Solution:

\((x - 1)\) ÷ \(\sqrt[2]{x} - 1\)

= \(\frac{(x - 1)}{\sqrt[2]{x} - 1}\)

= \(\frac{((\sqrt[2]{x})^{2} - 1^{2})}{\sqrt[2]{x} - 1}\)

= \(\frac{((\sqrt[2]{x} + 1)(\sqrt[2]{x} - 1)}{\sqrt[2]{x} - 1}\)….as we know \(a^{2} - b^{2} = (a + b)(a - b)\)

= \(\sqrt[2]{x}+1\).


8. Divide: √5 by √7

Solution:

√5 divided by √7

= √5 ÷ √7

= \(\sqrt{\frac{5}{7}}\)

= \(\frac{\sqrt{5}\times \sqrt{7}}{\sqrt{7}\times \sqrt{7}}\), [Rationalization of denominator of surds]

= √35/7.

 Surds






11 and 12 Grade Math

From Division 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. Expanded form of Decimal Fractions |How to Write a Decimal in Expanded

    Jul 22, 24 03:27 PM

    Expanded form of Decimal
    Decimal numbers can be expressed in expanded form using the place-value chart. In expanded form of decimal fractions we will learn how to read and write the decimal numbers. Note: When a decimal is mi…

    Read More

  2. Worksheet on Decimal Numbers | Decimals Number Concepts | Answers

    Jul 22, 24 02:41 PM

    Worksheet on Decimal Numbers
    Practice different types of math questions given in the worksheet on decimal numbers, these math problems will help the students to review decimals number concepts.

    Read More

  3. Decimal Place Value Chart |Tenths Place |Hundredths Place |Thousandths

    Jul 21, 24 02:14 PM

    Decimal place value chart
    Decimal place value chart are discussed here: The first place after the decimal is got by dividing the number by 10; it is called the tenths place.

    Read More

  4. Thousandths Place in Decimals | Decimal Place Value | Decimal Numbers

    Jul 20, 24 03:45 PM

    Thousandths Place in Decimals
    When we write a decimal number with three places, we are representing the thousandths place. Each part in the given figure represents one-thousandth of the whole. It is written as 1/1000. In the decim…

    Read More

  5. Hundredths Place in Decimals | Decimal Place Value | Decimal Number

    Jul 20, 24 02:30 PM

    Hundredths Place in Decimals
    When we write a decimal number with two places, we are representing the hundredths place. Let us take plane sheet which represents one whole. Now, we divide the sheet into 100 equal parts. Each part r…

    Read More