**Dividing Radicals**

The degrees of two radicals must be the same in order to divide them. For dividing two radicals, we use the quotient rule, which states that when two radicals of the same index are divided, the consequence is adequate to the radical of the division expression .

Where a, bel ∈ R, a ≥ 0, b-complex vitamin > 0, if n is even and north ≠ 0. bel ≠ 0, if n is odd .

While dividing two radicals, make a note that the denominator of the given expression is not a nothing. Remember that a negative radicand is permissible when the index of the group is negative. By using the division of radicals, we can write them in their simplify imprint. A free radical is said to be in its simplify kind if the denominator doesn ’ t have a extremist. so, rationalize the denominator if there is a radical in the denominator. To rationalize, we need to multiply both the numerator and denominator with the rationalizing factor. To understand the concept of systematization better let us consider an example. **Example: Simplify 4/(3 – 2√6).** **Solution: **

4/ ( 3 – 2√6 ) The rationalize factor is ( 3 + 2√6 ). now, multiply the numerator and denominator with the rationalizing component ( 3 + 2√6 ) = 4/ ( 3 – 2√6 ) × ( 3 + 2√6 ) / ( 3 + 2√6 ) = 4 ( 3 + 2√6 ) / ( 32 – ( 2√6 ) 2 ) { Since, ( a + bacillus ) ( a – bel ) = a2 – b2 } = 4 ( 3 + 2√6 ) / ( 9 – 24 ) = 4 ( 3 + 2√6 ) / ( -15 ) = -4 ( 3 + 2√6 ) /15 Hence, 4/ ( 3 – 2√6 ) = -4 ( 3 + 2√6 ) /15

**Sample Problems**

**Problem 1: Simplify 5√18/8√6.** **Solution:**

The given expression is 5√54/8√6 By using the quotient predominate, 5√18/8√6 = 5/8 × ( √18/√6 ) = 5/8 ( √ ( 18/6 ) = 5/8 × ( √3 ) = 5√3/8 Hence, 5√18/8√6 = 5√3/8 .

**Problem 2: Simplify ** **.** **Solution:**

The given expression is ³√56/ ³√7 By using the quotient principle, ³√56/ ³√7 =

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=

= ³√8 = ³√ ( 2 ) 3 = 2 Hence, ³√56/³√7 = 2 .

**Problem 3: Find the value of 5/(3+√7).** **Solution:**

5/ ( 3 + √7 ) immediately, multiply and divide the given term with ( 3 – √7 ) = 5/ ( 3 + √7 ) × ( 3 – √7 ) / ( 3- √7 ) = 5 ( 3 – √7 ) / ( 32 – 7 ) { Since, ( a + boron ) ( a – bel ) = a2 – b2 } = 5 ( 3 – √7 ) / ( 9 – 7 ) = 5 ( 3-√7 ) /2 Hence, 5/ ( 3 + √7 ) = 5 ( 3 – √7 ) /2

**Problem 4: Simplify ** . **Solution:**

By using the quotient rule, = = consequently, =

**Problem 5: Simplify √(72x2y3)/√(8y), if x > 0, y >0.** **Solution:**

√ ( 72x2y3 ) /√ ( 8y ) = = √ ( 9×2 ) = √ ( 3x ) 2 = 3x frankincense, √ ( 72x2y3 ) /√ ( 8y ) = 3x

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