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How to Simplify Square Roots (So It Finally Sticks)

You see a problem like 75\sqrt{75}75​ and your brain just goes, “Nope.”
The good news: simplifying square roots always follows the same small set of rules. Once you see the pattern, it becomes routine.

Quick Scoop

If you just want the core idea of how to simplify square roots :

  • Look for perfect squares inside the square root (like 4, 9, 16, 25, 36…).
  • Rewrite the number as (perfect square) × (leftover).
  • Take the square root of the perfect square and move it outside the radical.
  • Whatever isn’t a perfect square stays inside the radical.

Example:

75=25×3=53\sqrt{75}=\sqrt{25\times 3}=5\sqrt{3}75​=25×3​=53​

Here 25 is the perfect square, so its square root (5) comes out.

Mini-Section 1: What “Simplified Square Root” Really Means

A square root is simplified when:

  • There are no perfect-square factors left inside the root (except 1).
  • There are no fractions inside the radical that could be cleaned up.
  • There are no square roots in the denominator (once you learn rationalizing).

Some perfect squares you should know by heart:

  • 4,9,16,25,36,49,64,81,1004,9,16,25,36,49,64,81,1004,9,16,25,36,49,64,81,100 → their roots are 2,3,4,5,6,7,8,9,102,3,4,5,6,7,8,9,102,3,4,5,6,7,8,9,10.

Think of them as your “exit tickets” : whenever one shows up inside the radical, that part is allowed to leave.

Mini-Section 2: Two Main Methods (Prime Factors vs Perfect Squares)

There are two standard ways people learn how to simplify square roots :

Method 1: Perfect-Square Factor Method

This is the one most teachers prefer because it’s fast. Steps:

  1. Find the largest perfect square that divides the number inside the root.
  1. Rewrite as (perfect square)×(leftover)\sqrt{(\text{perfect square})\times (\text{leftover})}(perfect square)×(leftover)​.
  2. Take the square root of the perfect square out front.
  3. Leave the leftover inside.

Example 1: 75\sqrt{75}75​

  • Perfect square factor of 75: 25.
  • 75=25×3\sqrt{75}=\sqrt{25\times 3}75​=25×3​
  • 25=5\sqrt{25}=525​=5 → move it out: 535\sqrt{3}53​.

Example 2: 72\sqrt{72}72​

  • Perfect square factors of 72: 4, 9, 36; the largest is 36.
  • 72=36×2\sqrt{72}=\sqrt{36\times 2}72​=36×2​
  • 36=6\sqrt{36}=636​=6 → 626\sqrt{2}62​.

Method 2: Prime Factorization Method

This one is slower but bullet‑proof if you’re not sure about perfect squares.

Steps:

  1. Write the number as a product of prime factors.
  2. Pair identical factors inside the radical.
  3. For every pair, take one copy outside the radical.
  1. Multiply the outside numbers together; multiply the leftover inside numbers.

Example: 180\sqrt{180}180​

  • Prime factorization: 180=2×2×3×3×5180=2\times 2\times 3\times 3\times 5180=2×2×3×3×5.
  • Group pairs: (2×2)(2\times 2)(2×2), (3×3)(3\times 3)(3×3), and a lonely 5.
  • 180=22×32×5\sqrt{180}=\sqrt{2^2\times 3^2\times 5}180​=22×32×5​.
  • One 2 comes out, one 3 comes out: 2×35=652\times 3\sqrt{5}=6\sqrt{5}2×35​=65​.

Mini-Section 3: Step‑by‑Step Examples

These examples match what you’d see in most algebra resources explaining how to simplify square roots.

Example A: 50\sqrt{50}50​

  • 50=25×250=25\times 250=25×2.
  • 50=25×2=252=52\sqrt{50}=\sqrt{25\times 2}=\sqrt{25}\sqrt{2}=5\sqrt{2}50​=25×2​=25​2​=52​.

Example B: 20\sqrt{20}20​

  • 20=4×520=4\times 520=4×5.
  • 20=4×5=45=25\sqrt{20}=\sqrt{4\times 5}=\sqrt{4}\sqrt{5}=2\sqrt{5}20​=4×5​=4​5​=25​.

Example C: −27-\sqrt{27}−27​

  • Ignore the minus sign at first; simplify 27\sqrt{27}27​.
  • 27=9×327=9\times 327=9×3.
  • 27=93=33\sqrt{27}=\sqrt{9}\sqrt{3}=3\sqrt{3}27​=9​3​=33​.
  • Put the minus sign back: −33-3\sqrt{3}−33​.

Example D: A slightly harder one, 98\sqrt{98}98​

  • 98=49×298=49\times 298=49×2.
  • 98=492=72\sqrt{98}=\sqrt{49}\sqrt{2}=7\sqrt{2}98​=49​2​=72​.

Mini-Section 4: Fractions and the Quotient Rule

Once you’re comfortable with whole numbers, you’ll see square roots of fractions , like 964\sqrt{\frac{9}{64}}649​​.

There’s a helpful rule:

ab=abfor non‑negative a,b\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}\quad \text{for non‑negative }a,bba​​=b​a​​for non‑negative a,b

Example: 964\sqrt{\frac{9}{64}}649​​

  • Both 9 and 64 are perfect squares.
  • 964=964=38\sqrt{\frac{9}{64}}=\frac{\sqrt{9}}{\sqrt{64}}=\frac{3}{8}649​​=64​9​​=83​.

If the top or bottom isn’t a perfect square, you may:

  • Simplify the numerator and denominator separately.
  • Then, later in your course, learn to rationalize if there’s a root in the denominator.

Mini-Section 5: Combining Like Radicals

Once radicals are simplified, you can often add or subtract them if they match.

Think of them like like terms:

  • You can combine 23+53=732\sqrt{3}+5\sqrt{3}=7\sqrt{3}23​+53​=73​.
  • You cannot combine 22+532\sqrt{2}+5\sqrt{3}22​+53​ because the numbers inside (2 and 3) are different.

Sometimes the trick is to simplify first , then notice they match. Example: 212+932\sqrt{12}+9\sqrt{3}212​+93​

  • Simplify 12\sqrt{12}12​: 12=4×312=4\times 312=4×3, so 12=23\sqrt{12}=2\sqrt{3}12​=23​.
  • So 212=2×23=432\sqrt{12}=2\times 2\sqrt{3}=4\sqrt{3}212​=2×23​=43​.
  • Now the expression is 43+93=1334\sqrt{3}+9\sqrt{3}=13\sqrt{3}43​+93​=133​.

Mini-Section 6: Common Mistakes to Avoid

Even when people think they know how to simplify square roots , they fall into the same traps:

  • Writing a+b=a+b\sqrt{a+b}=\sqrt{a}+\sqrt{b}a+b​=a​+b​ (this is wrong in general).
  • Forgetting to simplify fully (leaving 20\sqrt{20}20​ instead of 252\sqrt{5}25​).
  • Not checking for bigger perfect squares (stopping at 4 instead of noticing 36, etc.).
  • Dropping the minus sign when you start with something like −27-\sqrt{27}−27​.

Mini-Section 7: Quick Practice Problems

Try these yourself using either method (perfect squares or prime factorization). Answers are in simplified radical form.

  1. 18\sqrt{18}18​ → 323\sqrt{2}32​.
  1. 45\sqrt{45}45​ → 353\sqrt{5}35​.
  1. 32\sqrt{32}32​ → 424\sqrt{2}42​.
  1. 48\sqrt{48}48​ → 434\sqrt{3}43​.
  1. 2536\sqrt{\frac{25}{36}}3625​​ → 56\frac{5}{6}65​.

Mini-Section 8: Forum‑Style Insight – Why It’s a Trending Topic

In school math help forums and Q&A sites, “how to simplify square roots” is one of the most asked algebra questions, especially around test season.

“I get the answer when I type it in a calculator, but not how teachers get 353\sqrt{5}35​ instead of 45\sqrt{45}45​.”

Learners today often rely on calculators, which give decimal answers like 6.7082…6.7082\ldots 6.7082… instead of simplified radicals like 353\sqrt{5}35​, but exams, especially in algebra and calculus, still expect the radical form. That’s why short guides and video tutorials showing simple steps on how to simplify square roots are consistently popular.

HTML Table: Perfect Squares Cheat Sheet

html

<table>
  <thead>
    <tr>
      <th>Perfect square</th>
      <th>Square root</th>
    </tr>
  </thead>
  <tbody>
    <tr><td>1</td><td>1</td></tr>
    <tr><td>4</td><td>2</td></tr>
    <tr><td>9</td><td>3</td></tr>
    <tr><td>16</td><td>4</td></tr>
    <tr><td>25</td><td>5</td></tr>
    <tr><td>36</td><td>6</td></tr>
    <tr><td>49</td><td>7</td></tr>
    <tr><td>64</td><td>8</td></tr>
    <tr><td>81</td><td>9</td></tr>
    <tr><td>100</td><td>10</td></tr>
  </tbody>
</table>

(These basic values show up constantly when you practice simplifying radicals.)

TL;DR Summary

  • How to simplify square roots : factor the number inside the radical, pull out perfect squares, leave the rest.
  • Use either the perfect‑square method or prime factorization ; both work, one is just faster.
  • Practice with common perfect squares so your eye instantly spots what to pull out of the root.

Bottom note: Information gathered from public forums or data available on the internet and portrayed here.