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# Simplifying Radicals – Techniques & Examples

The word radical in Latin and Greek means “**root**” and “**branch,**” respectively. The idea of radicals can be attributed to exponentiation or raising a number to a given power.

The concept of radical is mathematically represented as x ^{n}. This expression tells us that a number x is multiplied by itself n number of times. For instance,

3 ^{2} = 3 × 3 = 9, and 2 ^{4} = 2 × 2 × 2 × 2 = 16.

## How to Simplify Radicals?

A radical can be defined as a symbol that indicate the root of a number. Square root, cube root, fourth root are all radicals.*The following are the steps required for simplifying radicals:*

- Start by finding the prime factors of the number under the radical. Divide the number by prime factors such as 2, 3, 5 until only the left numbers are prime.
- Determine the index of the radical. The index of the radical tells the number of times you need to remove the number from inside to outside radical.
- Move only variables that make groups of 2 or 3 from inside to outside radicals.
- Simplify the expressions both inside and outside the radical by multiplying.
- Simplify by multiplication of all variables both inside and outside the radical.

*Example 1*

Simplify: √252

__Solution__

- Find the prime factors of the number inside the radical.

252 = 2 x 2 x 3 x 3 x 7

- Find the radical index, and for this case, our index is two because it is a square root. Therefore, we need two of a kind.

√ (2 x 2 x 3 x 3 x 7)

- Now pull each group of variables from inside to outside the radical. In this case, the pairs of 2 and 3 are moved outside.

2 x 3 √7

- By multiplication, simplify both the expression inside and outside the radical to get the final answer as:

6 √7

*Example 2*

Simplify:

^{3}√(-432x ^{7 }y ^{5})

__Solution__

- To solve such a problem, first, determine the prime factors of the number inside the radical.

432 = 2 x 2 x 2 x2 x 3 x 3 x 3

- Because, it is cube root, then our index is 3.

–^{3}√(2 x 2 x 2 x2 x 3 x 3 x 3 x x ^{7 }x y ^{5})

- Extract each group of variables from inside the radical, and these are 2, 3, x, and y.

-2 x 3 x y ^{3 }x x√(2xy ^{2})

- Multiply the variables both outside and inside the radical.

-6xy^{ 3}√(2xy ^{2})

*Example 3*

Solve the following radical problem.

Find the value of a number n if the square root of the sum of the number with 12 is 5.

__Solution__

- Write an expression of this problem, square root of the sum of n and 12 is 5

√(n + 12) = square root of the sum.

√(n + 12)=5

- Our equation which should be solved now, is:

√(n + 12) = 5

- On each side the equation is squared:

[√(n + 12)]² = 5²

[√(n + 12)] x [√(n + 12)] = 25

√[(n + 12) x √(n + 12)] = 25

√(n + 12)² = 25

n + 12 = 25

- Subtract 12 from both sides of the expression

n + 12 – 12 = 25 – 12

n + 0 = 25 – 12

n = 13