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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

## Practice Questions

*Multiply and simplify the following expressions:*

a. 3 √5 x − 4 √ 16

b. − 5√10 x √15

c. √12m x √15m

d. √5r ^{3} – 5√10r^{ 3}

*A kite is secured tied on the ground by a string. The wind blows such that the string is tight, and the kite is directly positioned on a 30 ft flag post. Find the height of the flag post if the length of the string is 110 ft long.*

*A school auditorium has 3136 total seats if the number of seats in the row is equal to the number of seats in the columns. Calculate the number total number of seats in a row.*

*The formula for calculating the speed of a wave is given as V=√9.8d, where d is the ocean’s depth in meters. Calculate the speed of the wave when the depth is 1500*

*A big squared playground is to be constructed in a city. Suppose the playground area is 400 and is to be subdivided into four equal zones for different sporting activities. How many zones can be put in one row of the playground without surpassing it?*