**Rationalization of surds :**

When the denominator of an expression contains a term with a square root or a number under radical sign, the process of converting into an equivalent expression whose denominator is a rational number is called rationalizing the denominator.

If the product of two irrational numbers is rational, then each one is called the rationalizing factor of the other.

**Case 1 :**

If the denominator is in the form of √a (where a is a rational number).

Then we have to multiply both the numerator and denominator by the same (√a).

**Example 1 :**

Rationalize the denominator 18/√6

**Solution :**

**Step 1: **

**We have to rationalize the denominator. Here we have ****√6 (in the form of ****√a). Then we have to multiply the numerator and denominator by ****√6 **

**Step 2 :**

By multiplying the numerators and denominators of first and second fraction , we get

**Step 3 :**

By simplifications, we get 3**√6**

**Case 2 :**

If the denominator is in the form of a ± √b or a ± c √b (where b is a rational number).

Then we have to multiply both the numerator and denominator by its conjugate.

a + √b and a - √b are conjugate of each other.

a + c√b and a - c√b are conjugate of each other.

**Example 2 :**

Rationalize the denominator 4/(1+2√3)

**Solution :**

**Step 1:**

**Here we have **(1+2√3)** (in the form of **a + c√b) in the denominator**. Then we have to multiply the numerator and denominator by the conjugate of **(1+2√3).

Conjugate of (1 + 2√3) is (1 - 2√3)** **

**Step 2 :**

**By multiplying the numerators and denominators of first and second fraction, we get**** **

**Step 3 :**

By comparing the denominator with the algebraic identity a² - b² = (a + b)(a - b), we get

**Case 3 :**

If the denominator is in the form of √a ± √b (where a and b are rational numbers).

Then we have to multiply both the numerator and denominator by its conjugate.

√a + √b and √a - √b are conjugate of each other.

**Example 3 :**

Rationalize the denominator (6 + √5)/(6-√5)

**Solution :**

**Step 1:**

**Here we have **(6-√5)** **in the denominator**. Then we have to multiply the numerator and denominator by the conjugate of **(6-√5).

Conjugate of (6-√5) is (6+√5)

**Example 4 :**

Rationalize the denominator (2 + √3)/(2 - √3) = x + y √3 and find the value of x and y.

**Solution :**

Now we have to compare the final answer with R.H.S

The values of x and y are 7 and 4 respectively.

- Rationalizing the denominator worksheet
- Logarithms
- Permutation and combination problems
- LCM and HCF word problems
- Time and distance word problems
- Mensuration
- Trigonometry
- Integration

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**