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# Multiplying Radicals – Techniques & Examples

A radical can be defined as a symbol that indicates the root of a number. Square root, cube root, fourth root are all radicals.

Mathematically, a radical is represented as x ^{n}. This expression tells us that a number x is multiplied by itself n number of times.

## How to Multiply Radicals?

For example, the multiplication of √a with √b is written as √a x √b. Similarly, the multiplication n ^{1/3} with y ^{1/2} is written as h ^{1/3}y ^{1/2}.

It advisable to place factors in the same radical sign. This is possible when the variables are simplified to a common index. For example, the multiplication of ^{n}√x with ^{n }√y is equal to ^{n}√(xy). This means that the root of several variables’ product is equal to the product of their roots.

*Example 1*

Multiply √8xb by √2xb.

__Solution__

√8xb by √2xb = √(16x ^{2 }b ^{2}) = 4xb.

You can notice that the multiplication of radical quantities results in rational quantities.

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*Example 2*

Find the product of √2 and √18.

__Solution__

√2 x √18 = √36 = 6.

### Multiplication of Quantities when the Radicands are of the Same Value

Roots of the same quantity can be multiplied by the addition of the fractional exponents. In general,

a ^{1/2} * a ^{1/3} = a ^{(1/2 + 1/3)} = a ^{5/6}

In this case, the denominator’s sum indicates the root of the quantity, whereas the numerator denotes how the root is to be repeated to produce the required product.

### Multiplication of Radical Quantities with Rational Coefficients

The radicals’ rational parts are multiplied, and their product is prefixed to the product of the radical quantities. For instance, a√b x c√d = ac √(bd).

*Example 3*

Find the following product:

√12x * √8xy

__Solution__

- Multiply all quantities the outside of radical and all quantities inside the radical.

√96x ^{2 }y

- Simplify the radicals

4x√6 y

*Example 4*

Solve the following radical expression

(3 + √5)/(3 – √5) + (3 – √5)/(3 + √5)

__Solution__

- Find the LCM to get,

[(3 +√5)² + (3-√5)²]/[(3+√5)(3-√5)]

- Expand (3 + √5) ² and (3 – √5) ² as,

3 ² + 2(3)(√5) + √5 ² and 3 ²- 2(3)(√5) + √5 ² respectively.

- Add the above two expansions to find the numerator,

3 ² + 2(3)(√5) + √5 ² + 3 ² – 2(3)(√5) + √5 ² = 18 + 10 = 28

- Compare the denominator (3-√5)(3+√5) with identity a ² – b ²= (a + b)(a – b), to get

3 ² – √5 ² = 4

- Write the final answer,

28/4 = 7

*Example 5*

Rationalize the denominator [(√5 – √7)/(√5 + √7)] – [(√5 + √7) / (√5 – √7)]

__Solution__

- By calculating the L.C.M, we get

(√5 – √7) ² – (√5 + √7) ² / (√5 + √7)(√5 – √7)

- Expansion of (√5 – √7) ²

= √5 ² + 2(√5)(√7) + √7²

- Expansion of (√5 + √7) ²

= √5 ² – 2(√5)(√7) + √7 ²

- Compare the denominator (√5 + √7)(√5 – √7) with the identity a² – b ² = (a + b)(a – b), to get,

√5 ² – √7 ² = -2

- Solve,

[{√5 ² + 2(√5)(√7) + √7²} – {√5 ² – 2(√5)(√7) + √7 ²}]/(-2)

= 2√35/(-2)

= -√35

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*Example 6*

Evaluate

(2 + √3)/(2 – √3)

__Solution__

- In this case, 2 – √3 is the denominator and rationalizes the denominator, both top, and bottom by its conjugate.

The conjugate of 2 – √3 is 2 + √3.

- Comparing the numerator (2 + √3) ² with the identity (a + b) ²= a ²+ 2ab + b ², the result is 2 ² + 2(2)√3 + √3² = (7 + 4√3).
- Comparing the denominator with the identity (a + b) (a – b) = a ² – b ², the results is 2² – √3².
- Answer = (7 + 4√3)

*Example 7*

Multiply √27/2 x √(1/108)

__Solution__

√27/2 x √(1/108)

= √27/√4 x √(1/108)

= √(27 / 4) x √(1/108)

= √(27 / 4) x √(1/108) = √(27 / 4 x 1/108)

= √(27 / 4 x 108)

Since 108 = 9 x 12 and 27 = 3 x 9

√(3 x 9/ 4 x 9 x 12)

9 is a factor of 9, and so simplify,

√(3 / 4 x 12)

= √(3 / 4 x 3 x 4)

= √(1 / 4 x 4)

=√(1 / 4 x 4) = 1 / 4