Adding and Subtracting Radicals Expressions

Adding and subtracting radicals involves combining radical expressions with the same index and radicand. In this article, let's learn about the addition and subtraction of square roots in detail.

What is Radical?

As, square root and square are inverse operations, radical is the inverse operation of the exponents function. Radical is an expression that has a root, mostly a square root.

For example, √(36) is radical and its value is,

√(36) = √(6×6) = 6

Now coming to the addition and subtraction of square roots, we can perform the operations just like we do with regular numbers. But remember that we can only add or subtract square roots or radicals that have the same radicand.

How to Add and Subtract Radicals?

We can only add or subtract square roots or radicals that have the same radicand. If two terms have the same radicand, then we can add or subtract their coefficients and leave the radicand as it is. The terms that have the same radicands are known as "like radicals", whereas the terms that have different radicands are known as "unlike radicals."

Steps to Add or Subtract Radicals

Follow the steps added below to add or subtract radicals.

Step 1: Simplify the given square roots if possible. So, try to factor them to find at least one perfect square factor.

Step 2: Once you have simplified the given square roots of the terms, find the like radicals.

Step 3: Finally, add or subtract the coefficients of like radicals and leave any additional terms as part of the equation.

This is explained by the example added below:

Example: Solve 8√9 + 3√16.

Solution:

Given expression is 8√9 + 3√16

Here, both radicands are different. But we can simplify them further

8√9 + 3√16 = 8√(3²) + 3√(4²)

= 8 × 3 + 3 × 4

= 24 + 12 = 36.

Thus, 8√9 + 3√16 = 36.

Also, Check

• Algebraic Expressions
• Types of algebraic expressions

Solved Examples

Example 1: Simplify: 5√8 + 3√32.

Solution:

Given expression: 5√8 + 3√32

Now, simplifying radicals, we get

5√(2×2×2) + 3√(2×2⁴)

= 5 × 2√2 + 3 × 2²√2

= 10√2 + 12√2

= 22√2

Hence, 5√8 + 3√32 = 22√2.

Example 2: Simplify 14√3 - 2√12.

Solution:

Given Expression: 14√3 - 2√12

Now, simplifying radicals, we get

14√3 - 2√12

= 14√3 - 2√(2²×3)

= 14√3 - 2 ×2√3

= 14√3 - 4√3  = 10√3

Hence, 14√3 - 2√12 = 10√3.

Example 3: Solve: 7√(a²) - 2√(a⁴) + √(a²).

Solution:

Given Expression: 7√(a²) - 2√(a⁴) + √(a²)

Now, simplifying radicals, we get

= 7a - 2a² + a

= 8a - 2a²

Thus, 7√(a²) - 2√(a⁴) + √(a²) = 8a - 2a²

Example 4: Solve: 51√7 + 16√5 - 13√7 + 31√5.

Solution:

Given Expression: 51√7 + 16√5 - 13√7 + 31√5

= (51√7 - 13√7) + (16√5 + 31√5)

= 38√7 + 47√5

Thus, 51√7 + 16√5 - 13√7 + 31√5 = 38√7 + 47√5.

Example 5: Solve: 19√75 + 12√27 - 10√48.

Solution:

Given Expression: 19√75 + 12√27 - 10√48

= 19√(3 × 5 × 5) + 12√(3 × 3 × 3) - 10√(4 × 4 × 3)

= 19 × 5√3 + 12 × 3√3 - 10 × 4√3

= 95√3 + 36√3 - 40√3

= 91√3

Thus, 19√75 + 12√27 - 10√48 = 91√3.

Example 6: Simplify: 2√3 + 3√3.

Solution:

Given expression: 2√3 + 3√3

Now, simplifying radicals, we get

= 2√3 + 3√3

= 5√3

Hence, 2√3 + 3√3 = 5√3

Example 7: Simplify: √9 + √25.

Solution:

Given expression: √9 + √25

Now, simplifying radicals, we get

= √9 + √25

= 3 + 5 = 8

Hence, √9 + √25 = 8

Example 8: Simplify: √8 + 2√2

Solution:

Given expression: √8 + 2√2

Now, simplifying radicals, we get

= √8 + 2√2

= 2√2 + 2√2          ( As √8 = 2√2 )

= 4√2

Hence, √8 + 2√2 = 4√2

Example 9: Simplify: √3×(4√3 + 11 )

Solution:

Given expression: √3×(4√3 + 11 )

Now, simplifying radicals, we get

= √3×(4√3 + 11 )

= √3×4√3 + 11×√3

= 4×3 + 11×√3

= 12 + 11×√3

Example 10: Simplify: √5×(√5 + √6 )

Solution:

Given expression: √5×(√5 + √6 )

Now, simplifying radicals, we get

= √5×(√5 + √6 )

= √5×√5 + √5×√6

= 25 + √30

What is a square root?

Square root of any number is a value that gives the original number when multiplied by itself. The square root and square are inverse operations. For example, if a number "m" is the square root of a number "n"(m = √n), then the "n" is the square of "m" (n = m × m).

Can square root be negative?

Square root of a number can be both positive or negative. For example, the value of the square root of 9 is equal to 3 and -3.

What is the symbol of a square root?

Square root of x is denoted as √x, where x is called a radicand and "√" is called the radical symbol, which denotes the square root.

How to add and subtract square roots?

Addition and subtraction of square roots can be performed just like we do with regular numbers. But remember that we can only add or subtract square roots or radicals that have the same radicand.