Radical expression

Radical expression

Any mathematical expression containing a radical symbol(√) is called a radical expression. Generally we use the symbol '√' 

to determine the square root of a number. But, this symbol may be used to represent the 'cube' , 'fourth', or higher roots of a number.

Definition:

In mathematics, any expression containing the radical symbol (√) is known as radical expression. 

Let's explain the matter...

  Let, 'n' be any positive integer (n>1) and 'a' be any real number; then the expression, ⁿ√a or  (a)ˡ/ⁿ  is called a radical expression. Here, 'a' is known as radicand and the symbol '√' is known as radical. Here, the form  (a)ˡ/ⁿ  has a special name: "exponent form". 

  Examples:

   √16=4=(16)ˡ/² , √5,√7,...etc.

History:

The term "radical" is derived from a Latin word 'radix'. In Latin 'radix' means 'root'. 

In 1600s radical expressions were first used in England. Then the uses of radical expression spread worldwide.

Properties of radicals:

If a(>0) and b(>0) , then

(1) √a×√b = √(a×b)

(2)√(a/b)=(√a)/(√b)

(3) √(a+b) is not equals to (√a+√b).

(4) √(a-b) is not equals to (√a-√b).

Note:

  The expression (√a+√b) is called the conjugate of the expression (√a-√b). Therefore, the expressions (√a+√b) and (√a-√b) are conjugate to each other. so, we can use the conjugate to rationalize the denominator of a radical expression.

Simplified radical expression:

A radical expression is said to be in simplified radical form,  if each of the following are true:

(1) All exponents in the radicand must be less than the index.

(2) Any exponents in the radicand can have no factors in common with the index.

(3)No fractions appear under a radical.

(4) No radicals appear in the denominator of a fraction.

   As for example, simplyfy: ⁹√(a⁶).

  We have , ⁹√(a⁶) = (a⁶)ˡ/⁹ = (a)⁶/⁹= (a)²/³= ³√a².

  So, simplified radical expression of ⁹√a⁶ is ³√a².

Method to simplyfy a radical expression:

There are many radical expressions , where the radicand is not a perfect square or cubes or higher powers of a number. In such cases to simplyfy the expression , we may use the following method.

  At first factories the radicand in all possible prime factors. Then collect the same prime factors and move them outside the radical sign. Then multiply the factors inside and outside the radical sign separately. The result is in the simplified form.

As for example, simplyfy: √480.

  we have, √ 480=√(2×2×2×2×3×2×5)

                             =2×2√(3×2×5)=4√30.

Some formulas to solve a radical expression:

(1) (aᵐ)×(aⁿ)=aᵐ⁺ⁿ 

(2) (aᵐ)÷(aⁿ)=aᵐ⁻ⁿ

(3) (aᵐ)ⁿ= aᵐⁿ

(4) (ab)ᵐ= (aᵐ)×(bᵐ)

(5)(a÷b)ᵐ= (aᵐ)÷(bᵐ)

(6) If m is a positive number, a⁻ᵐ=1÷(aᵐ). Here, a⁻ᵐ is called reciprocal of aᵐ.

(7) If m,n are integer, aᵐ/ⁿ means (aᵐ)ˡ/ⁿ ; i.e., n-th root of aᵐ.

(8) If m=0, a⁰ is meaningless, a⁰=1.

(9) If a,m,n real and aᵐ=aⁿ then, m=n, where, a not equal to 0,1,-1.

(10) If a, b, m are real , and aᵐ= bᵐ then, either a=b or m=0.

If you find out any incorrect information or know anything more about this , please write it in the comment section!