How to divide Radicals? – GeeksforGeeks

In mathematics, a root is referred to as an expression involving a root. For model, the squarely root and cube roots are common radicals that are represented as √ and ³√, respectively. A root is considered to be a square root if its index is not mentioned. For example, “ nth root of ( ten + 2y ) ” is symbolically written as shown in the figure given below, where n is called the index of the extremist and ( x + 2y ) is the radicand. It will be a cube root if the respect of n is 3 and a square root of north is 2. There are sealed things we should remember while using revolutionary expressions. The index “ n ” of a radical is a positive number that is greater than “ 2 ”. The radicand of a radical must be a substantial total. If the index of a revolutionary is even, the radicand must be a positive issue greater than or equal to zero. For case, if a radicand is a damaging number and the index is evening, then the leave will be an irrational count. If an index is an leftover count and the radicand is positivist, then the result is positive. If the radicand is negative, then the results will besides be negative .

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 .

\frac{\sqrt[n]{a}}{\sqrt[n]{b}}= \sqrt[n]{\frac{a}{b}}

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  \frac{\sqrt[3]{56}}{\sqrt[3]{7}}   . Solution:

The given expression is ³√56/ ³√7 By using the quotient principle, ³√56/ ³√7 =  \sqrt[3]{\frac{56}{7}}

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=   \sqrt[3]{\frac{7\times8}{7}} = 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  \sqrt[5]{\frac{39}{32}} . Solution:

By using the quotient rule, \sqrt[5]{\frac{39}{32}} = \sqrt[5]{\frac{39}{2^{5}}} = \frac{\sqrt[5]{39}}{2} consequently, \sqrt[5]{\frac{39}{32}}  =

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

√ ( 72x2y3 ) /√ ( 8y ) = \sqrt{\frac{72x^{2}y^{3}}{8y}} = √ ( 9×2 ) = √ ( 3x ) 2 = 3x frankincense, √ ( 72x2y3 ) /√ ( 8y ) = 3x

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