How to divide Radicals?
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 β and Β³β, respectively. A radical is considered to be a square root if its index is not mentioned. For example, βnth root of (x + 2y)β 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 βnβ of a radical is a positive number that is greater than β2β. 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.
Where a, b β R, a β₯ 0, b > 0, if n is even and n β 0.
b β 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 doesnβt 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 β 2β6).
Solution:
4/(3 β 2β6)
The rationalizing factor is (3 + 2β6). Now, multiply the numerator and denominator with the rationalizing factor (3 + 2β6)
= 4/(3 β 2β6) Γ (3 + 2β6)/(3 + 2β6)
= 4(3 + 2β6)/(32 β (2β6)2) {Since, (a + b)(a β b) = a2 β b2}
= 4(3 + 2β6)/(9 β 24)
= 4(3 + 2β6)/(-15)
= -4(3 + 2β6)/15
Hence, 4/(3 β 2β6) = -4(3 + 2β6)/15
Sample Problems
Problem 1: Simplify 5β18/8β6.
Solution:
The given expression is 5β54/8β6
By using the quotient rule,
5β18/8β6
= 5/8 Γ (β18/β6)
= 5/8 (β(18/6)
= 5/8 Γ (β3)
= 5β3/8
Hence, 5β18/8β6 = 5β3/8.
Problem 2: Simplify .
Solution:
The given expression is Β³β56/ Β³β7
By using the quotient rule,
Β³β56/ Β³β7
=
=
= Β³β8 = Β³β(2)3
= 2
Hence, Β³β56/Β³β7 = 2.
Problem 3: Find the value of 5/(3+β7).
Solution:
5/(3 + β7)
Now, multiply and divide the given term with (3 β β7)
= 5/(3 + β7) Γ (3 β β7)/(3- β7)
= 5(3 β β7)/(32 β 7) {Since, (a + b)(a β b) = a2 β b2}
= 5(3 β β7)/(9 β 7)
= 5(3-β7)/2
Hence, 5/(3 + β7) = 5(3 β β7)/2
Problem 4: Simplify .
Solution:
By using the quotient rule,
=
=
Therefore, =
Problem 5: Simplify β(72x2y3)/β(8y), if x > 0, y >0.
Solution:
β(72x2y3)/β(8y)
=
= β(9x2)
= β(3x)2
= 3x
Thus, β(72x2y3)/β(8y) = 3x
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