Square Root of 180 Simplified

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What is the square root of 180 simplified?

Answer

√180 = 6√5

180 = 36 × 5, and 36 = 6² is the largest perfect square that divides 180, so √180 = √36 × √5 = 6√5. Check: 6√5 ≈ 13.416, the same as √180 ≈ 13.416.

Simplifying √180 Step by Step

1. Prime factorization. — All Prime Factors of 180:

180 = 2 × 2 × 3 × 3 × 5 = 22 × 32 × 5

2. Pair up equal primes. Each pair is a perfect square and comes out of the root as one factor; a prime without a partner stays under the radical.

Prime factorTimes in 180Comes out of the rootStays under the root
222—
323—
51—5

3. Take the pairs out of the root.

√180 = √((2 × 2) × (3 × 3) × 5)

= 2 × 3 × √5

= 6√5

4. Check by squaring back.

6² × 5 = 36 × 5 = 180

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This calculator writes a square root in simplest radical form: it finds the largest perfect square that divides the number, takes its root out of the radical and leaves the rest under it. For example, it can help you find out what is the square root of 180 simplified? (The answer is: 6√5). Enter a whole number (e.g. '180') and hit the 'Simplify' button.
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Simplified Square Roots Table

NumberSimplest radical formDecimal value
165√165≈ 12.845
166√166≈ 12.884
167√167≈ 12.923
1682√42≈ 12.961
1691313
170√170≈ 13.038
1713√19≈ 13.077
1722√43≈ 13.115
173√173≈ 13.153
174√174≈ 13.191
1755√7≈ 13.229
1764√11≈ 13.266
177√177≈ 13.304
178√178≈ 13.342
179√179≈ 13.379
1806√5≈ 13.416
181√181≈ 13.454
182√182≈ 13.491
183√183≈ 13.528
1842√46≈ 13.565
185√185≈ 13.601
186√186≈ 13.638
187√187≈ 13.675
1882√47≈ 13.711
1893√21≈ 13.748
190√190≈ 13.784
191√191≈ 13.82
1928√3≈ 13.856
193√193≈ 13.892
194√194≈ 13.928

FAQ

What is the square root of 180 simplified?

√180 = 6√5

Is 180 a perfect square?

No. 180 = 36 × 5: its largest perfect square factor is 36 = 6², and the factor 5 stays under the radical, so √180 = 6√5.

What is √180 as a decimal?

√180 ≈ 13.416. The decimal never ends and never repeats, so it is always rounded; the exact value is 6√5.

What is the largest perfect square that divides 180?

36 = 6². Dividing gives 180 ÷ 36 = 5, and 5 has no square factor left, so √180 = √36 × √5 = 6√5.

See Also