How to simplify radicals ?

The radical or root is the mathematical expression opposite of the exponent just like addition is the opposite of subtraction. The smallest radical is known as square root and it’s represented with symbol √. Next radical is cube root and it’s represented with symbol ³√.  Do you how to simplify the radicals?

The number in front of radical is called it’s index number. Index number could be a whole number and also signifies the exponent and is used to cancel out the radical. For instance, to rise to power 3, it would cancel out the cube root. An exponent is the number of times the number gets multiplied with itself.

Example:  3³ = 3*3*3=27

What is Radicals?

Key Terms:

1. Radical
2. Square root
3. Perfect square
4. Cube root

Aims:

1. To know the meaning of the radical symbol
2. To know the square roots and relatively simple high-order roots
3. Relating roots to the fractional exponent
4. Applying the rule of the exponent to derive rules for roots of products & quotients

Now, we will analyze the actual meaning of the fractional exponent and will relate them to the radicals. The radical is the symbol and also refers to the specific root of a number. The symbol is “√”.

The radical in itself represents square root. The square root of a number n can be written as: √n

Let the square root of N is taken as another number R. The square or second power of R is represented symbolically as:

√N = R

⇒ N= R²……………………………………………………………………………………………………………………………… (1)

To clarify more, for example consider the number 25. The square root of 25 is 5. It is represented symbolically as:

25=5*5=5²………………………………………………………………………………………………………………………….. (2)

√25 = 5………………………………………………………………………………………………………………………………. (3)

Similarly other numbers like 1, 4, 9, 16, are perfect squares.

In case fractional values, they will also have square roots-

For example, ¼=1/2*1/2 = (1/2)²

⇒ √1/4 = ½   …………………………………………………………………………………………………………………………(4)

We can also calculate the square root of “0”.

√0 = 0

⇒ 0² =0*0 ………………………………………………………………………………………………………………………… (5)

This means we can calculate the square root of a number that is greater than or equal to zero.

But, we can’t calculate the square root of a negative number. It is invalid.

However, the product of two negative numbers is positive. This can be represented as follows:

For instance, (-4) * (-4) = 16

Similarly, we can also calculate the third root or cube root of a number. Let us consider a number N and its cube root is R. Now, it’s represented as:

³√N =R

⇒ N = R³………………………………………………………………………………………………………………………………. (6)

For example, ³√125 = 5 ……………………………………………………………………………………………………… (7)

Now ^{we w}ill know about the calculation of the K^th root of a number N.  Let the K^th root of a number is

^K√N = R

⇒ N = R^K ……………………………………………………………………………………………………………………………… (8)

For example, 256 = 4*4*4*4 = 4⁴

⁴√256 =4

Few more examples,   2187 = 3*3*3*3*3*3*3 =3⁷

⁷√2187 = 3 …………………………………………………………………………………………………………………………. (9)

Fractional Exponents

Take the following example-

√N = R

⇒ N= R²

Take following expression-

(N^a) ^b = N^a*b = N^ab ………………………………………………………………………………………………………………… (10)

Now, we will express the above expression in numerical form-

N = (N^1/2)² = (R)² =R²  …………………………………………………………………………………………………………. (11)

⇒ √N = N^1/2 = R   ……………………………………………………………………………………………………………… (12)

⇒ N = R^1/2 ………………………………………………………………………………………………………….. (13)

Now we can similarly calculate the K^th root of the number N and is equal to N is raised to the power of 1/K.

^K√N = N^1/k ……………………………………………………………………………………………………………………… (14)

For example, √9/25 = √9/√25 = 3/5

27^2/3 = (27^1/3)² = (³√27)² = (3)² =9 …………………………………………………………………………………..(15)