How to Get Rid of a Square Root- Simplification Techniques

What "Getting Rid of a Square Root" Actually Means

When math teachers say "get rid of the square root," they don't mean the root disappears. It moves. It changes form. The goal is to rewrite an expression so there's no radical in the denominator or to simplify the expression into something cleaner.

Here's the bitter truth: there's no magic button. You need to understand three core techniques and know when to use each one.

The Three Techniques You Need to Know

1. Simplify the Radical First

Before doing anything else, check if the number under the square root has a perfect square factor. This is always your first step.

Take √50. The number 50 = 25 × 2. Since 25 is a perfect square, you can pull it out:

√50 = √(25 × 2) = √25 × √2 = 5√2

That's it. You didn't eliminate the root—you made it smaller and simpler.

2. Rationalize the Denominator

This is the technique your teacher really cares about. If you have a radical in the denominator, multiply both the top and bottom by the conjugate of the denominator.

For example:

1/√3

Multiply by √3/√3:

(1 × √3) / (√3 × √3) = √3/3

The root moved from the bottom to the top. The expression is now rationalized.

For a two-term denominator like 1/(√5 + 2), use the conjugate (√5 - 2):

1 × (√5 - 2) / (√5 + 2)(√5 - 2) = (√5 - 2) / (5 - 4) = √5 - 2

This works because (a + b)(a - b) = a² - b². The radicals cancel out.

3. Isolate and Square Both Sides

This one is for equations, not expressions. If you have √x = something, square both sides to remove the root.

√x = 5

Squaring both sides: (√x)² = 5²

This gives you x = 25

Always check your answer. Squaring can introduce fake solutions called extraneous roots.

Quick Reference Table

Getting Started: Step-by-Step

Here's how to approach any radical problem:

• Step 1: Simplify the radical if possible. Look for perfect square factors.
• Step 2: Identify where the radical is located—in the numerator or denominator.
• Step 3: If it's in the denominator, rationalize it. Multiply by the radical or the conjugate.
• Step 4: If you're solving an equation, isolate the radical first, then square both sides.
• Step 5: Check your work. Plug answers back in to verify.

Common Mistakes That Will Cost You Points

Students consistently mess up in three ways:

Splitting radicals incorrectly: √(a + b) ≠ √a + √b. This is wrong. You can only split multiplication, not addition.

Forgetting to multiply both terms: When rationalizing 1/(√3 + 1), you must multiply both terms in the numerator by the conjugate. Not just one.

Not simplifying completely: √98 should become 7√2, not left as √98. Partial credit isn't a thing on most tests.

Practice Makes It Automatic

You won't get this from reading. You need to work through 20-30 problems before it clicks. Here's a mini problem set:

• Simplify: √75
• Rationalize: 5/√2
• Rationalize: 2/(√6 - 1)
• Solve: √(x + 1) = 4

Answers: 5√3, 5√2/2, (√6 + 1)/5, x = 15