Rationalization

**Category : **9th Class

The process of making denominator of a irrational number to a rational by multiplying with a suitable number is called rationalization. This process is adopted when the denominator of a given number is irrational. The number by which we multiply the denominator or convert it into rational is called rationalizing factor.

**Rationalize the denominator of \[\frac{6}{\sqrt{7}+\sqrt{2}}\].**

**Solution:**

We have:

\[\frac{6}{\sqrt{7}+\sqrt{2}}=\frac{6\times (\sqrt{7}-\sqrt{2})}{(\sqrt{7}+\sqrt{2})(\sqrt{7}-\sqrt{2})}=\frac{6\times (\sqrt{7}-\sqrt{2})}{7-2}\]

Here, \[(\sqrt{7}-\sqrt{2})\] is rationalizing factor.

Therefore, \[\frac{6}{\sqrt{7}+\sqrt{2}}=\frac{6}{5}(\sqrt{7}-\sqrt{2})\]

** Laws of Radicals Let**

\[x>0\] be any real number if a and b rational number then

(i) \[({{x}^{a}}\times {{x}^{b}})={{x}^{a+b}}\]

(ii) \[{{({{x}^{a}})}^{b}}={{x}^{ab}}\]

(iii) \[\frac{{{x}^{a}}}{{{x}^{b}}}={{x}^{a-b}}\]

(iv) \[{{x}^{a}}\times {{y}^{a}}={{(xy)}^{a}}\]

** Euclid's Division Lemma**

Let a and b be any two positive integer. Then, there exist unique integers q and r such that \[a=\text{bq}+\text{r},\text{ }0\le \text{r}<\text{b}\]

- \[1\times 9+2=11\] \[9\times 9+7=88\] \[9\times 9=81\] \[6\times 7=42\]
- \[12\times 9+3=111\] \[98\times 9+6=888\] \[99\times 99=980166\times 67=4422\]
- \[123\times 9+4=1111\] \[987\times 9+5=8888\] \[999\times 999=998001\] \[666\times 667=444222\]

- A particular point on the number line represents a particular rational number.
- A rational number cannot be represented by two or more than two distinct points on a number line.
- There are infinite real numbers between two distinct real numbers.
- A and B be two rational number in which A < B then the n rational number between A and B are \[(A+x),(A+2x),........(A+nx)\] Where \[x=\frac{B-A}{n+1}\]
- A non-terminating and non-repeating decimals are called irrational numbers.
- The sum of two irrational numbers may or may not be irrational.
- The difference of two rational numbers may or may not be irrational.
- The quotient of two irrational number may not be irrational.
- The quotient of two irrational number may or not be irrational.

**Which one of the following statements is true for a rational number?**

(a) It is in the form of \[\frac{p}{q}\], Where p and q are integers and\[p\ne 0\]

(b) The decimal representation of a rational number is either terminating or non-terminating and non-repeating decimals

(c) There are five rational number between two given rational numbers

(d) Rational numbers are either terminating or non-terminating and repeating decimals

(e) None of these

**Answer:** (d)

** Match the following:**

(a) \[\frac{123}{128}\] | (i) Non terminating and repeating decimal |

(b) \[\frac{2318}{9900}\] | (ii) Non terminating and non-repeating decimals. |

(c) 0.01010010010001000... | (iii) Rational number between two rational numbers\[x\]and y. |

(d) \[\frac{1}{2}(x+y)\] | (iv) Terminating decimals. |

(a) a-iv, b-i, c-ii, d-iii

(b) a-ii, b-i, c-iv, d-iii

(c) a-iv, b-iii, c-ii, d-i

(d) a-ii, b-iv, c-iii, d-i

(e) None of these

**Answer:** (a)

**Explanation:**

(a) \[\frac{123}{128}\] Here, denominator is 128 whose prime factor \[~\text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\times \text{2}\] have only 2 as a factor therefore, it is a terminating decimal, that is why (iv) is correct for (a).

(b) \[\frac{2318}{9900}\] the denominator is 9900 whose prime factor is \[\text{2}\times \text{3}\times \text{3}\times \text{5}\times \text{11}\]. Here, 3 and 11 as a factor which is other than 2 and 5, that is why \[\frac{2318}{9900}\] is non-terminating and repeating decimal, (b-i)

(c) 0.01010010010001000......... Here, number on the right of the decimal is not repeating periodically, that is why it is non-terminating and non - repeating decimal, (c-ii)

(d) For any two rational numbers \[x\] and y, the rational number which is between \[x\] and y is \[\frac{1}{2}(x+y).(d-iii)\].

**Read the following statements.**

(i) Rational number may or may not be an integer

(ii) Some rational number can be represented on a number line

(iii) On a number line only rational numbers can be represented

(iv) There are infinite number of rational number between two given rational numbers

**Which one of the following set of statements is correct?**

(a) (i) and (ii)

(b) (i) and (iv)

(c) (i), (iii) and (iv)

(d) (i), (ii) and (iv)

(e) None of these

**Answer:** (b)

**Which one of the following is repeating decimals?**

(a) \[\frac{22}{7}\]

(b) \[\pi \]

(c) \[\frac{224}{135}\]

(d) \[\frac{154}{448}\]

(e) None of these

**Answer:** (d)

**The length, breadth and height of a room are 5 m 25 cm, 3 m 25 cm and 1 m 25 cm respectively. The length of the longest rod which can measure the three dimensions of the room exactly will be:**

(a) 50cm

(b) 75cm

(c) 1m

(d) 25cm

(e) None of these

**Answer:** (d)

*play_arrow*Introduction*play_arrow*Decimal Representation of Numbers*play_arrow*Irrational Number*play_arrow*Real Number*play_arrow*Rationalization*play_arrow*Real Numbers*play_arrow*Real Numbers

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