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How to divide Radicals?

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In mathematics, a radical is referred to as an expression involving a root. For example, the square root and cube roots are common radicals that are represented as √ and ³√, respectively. A radical is considered to be a square root if its index is not mentioned. For example, “nth root of (x + 2y)” is symbolically written as shown in the figure given below, where n is called the index of the radical and (x + 2y) is the radicand. It will be a cube root if the value of n is 3 and a square root of n is 2. There are certain things we should remember while using radical expressions. The index “n” of a radical is a positive number that is greater than “2”. The radicand of a radical must be a real number. If the index of a radical is even, the radicand must be a positive number greater than or equal to zero. For instance, if a radicand is a negative number and the index is even, then the result will be an irrational number. If an index is an odd number and the radicand is positive, then the result is positive. If the radicand is negative, then the results will also 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 result is equal to the radical of the division expression. 

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

Where a, b ∈ R, a ≥ 0, b > 0,  if n is even and n ≠ 0.

b ≠ 0, if n is odd.

While dividing two radicals, make a note that the denominator of the given expression is not a zero. Remember that a negative radicand is allowable when the index of the radical is negative. By using the division of radicals, we can write them in their simplified form. A radical is said to be in its simplified form if the denominator doesn’t have a radical. 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 rationalization better let us consider an example.

Example: Simplify 4/(3 – 2√6).

Solution: 

4/(3 – 2√6)

The rationalizing factor is (3 + 2√6). Now, multiply the numerator and denominator with the rationalizing factor (3 + 2√6)

= 4/(3 – 2√6) × (3 + 2√6)/(3 + 2√6)

= 4(3 + 2√6)/(32 – (2√6)2)    {Since, (a + b)(a – b) = 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 rule,

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 rule,

³√56/ ³√7

= \sqrt[3]{\frac{56}{7}}

= \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)

Now, multiply and divide the given term with (3 – √7)

= 5/(3 + √7) × (3 – √7)/(3- √7)

= 5(3 – √7)/(32 – 7)     {Since, (a + b)(a – b) = 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}

Therefore, \sqrt[5]{\frac{39}{32}}   = \frac{\sqrt[5]{39}}{2}

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

Solution:

√(72x2y3)/√(8y)

\sqrt{\frac{72x^{2}y^{3}}{8y}}

= √(9x2)

= √(3x)2

= 3x

Thus, √(72x2y3)/√(8y) = 3x



Last Updated : 02 Jul, 2022
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