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**Algebraic Expressions:** An algebraic expression has at least one variable and is connected by a single arithmetic operator. Addition (+), subtraction (-), multiplication (*), and division (/) are examples of these operators.

## A Brief Outline

The square root or square of algebraic expressions is itself an algebraic expression, and the same is true for arithmetic expressions.

**Algebraic Expressions: Finding the Square Root**

There are two ways to solve the square root of algebraic expressions:

- The Factoring Technique
- The Method of Division

**Important Concepts**

**Finding Square Root by Division Method Examples**

The expression is specified as 36x^{4} – 36x^{2} + 9

Now, the maximum power of the variable, x, is 4.

So, the square root of the above equation will be x^{2}

Because √x^{4} = x^{2}, Now the coefficient is x^{2 }is 36. So, let us divide first as 6x^{2}

As 6x^{2 }is the only term and by multiplying by 2, we get 12x^{2}

Accordingly, the first term to be divided will be -36x^{2,} and on dividing -36x^{2} by 12x^{2,} we get -3.

We have to multiply 12x^{2}-3 by 3 to get -36x^{2}+ 9

The remainder is crooked to zero, and the given expression wholly divisible by 6x^{2}-3

So, the square root of 36x^{4} – 36x^{2} + 9 will be 6x^{2} – 3

**Finding Square Root by Factoring Method Examples**

The factor theorem asserts that if an algebraic expression f(x) is divisible by x=p with a zero remainder, then f(p) = 0 or (x – p) is a factor of f(x) (x).

This theorem is the inverse of the remainder theorem, statings that if f(x) is divisible by x-p, then perhaps the remainder is f. (p).

**Illustration**:

Find the square root of (b+1/b)^{2} + 4(b+1/b) + 4

**Solution**: Contemplate (b+1/b) as a.

Replacing the value of “a” in the given expression, we get a^{2} + 4a + 4.

The given expression is in the form: (a + 2)^{2} and one of its factor is (a + 2)

So, the square root of the specified expression is (b+1/b) + 2.

## Significance of how to find the square root of algebraic expressions in the IIT JEE exam

This unit comprises most of the class XI chapters (plus one chapter from the 12th). The weighted average of these maths chapters is given over the JEE Main syllabus. Algebra and Calculus are the two high-weighted units. The algebra chapter provides for 35% of the entire mathematics curriculum.

#### FAQs

##### What is the definition of square root?

The square root of a number is the factor of a number that produces the original number when multiplied by itself. Exceptional exponents comprise squares & square roots.

##### What Is the Best Way to Minimize Square Root?

We must first determine the number's prime factorization to minimise a square root. Keep it under the square root sign if you can't group a factor.

**Question: How do I determine the square root of a number?**

**Answer:**

- Square Root Method with Repeated Subtraction
- Prime Factorization Method for Square Root
- Estimation Method for Square Root
- Long Division Method for Square Root