Square Root of 224 Simplified

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What is the square root of 224 simplified?

Answer

√224 = 4√14

224 = 16 × 14, and 16 = 4² is the largest perfect square that divides 224, so √224 = √16 × √14 = 4√14. Check: 4√14 ≈ 14.967, the same as √224 ≈ 14.967.

Simplifying √224 Step by Step

1. Prime factorization. — All Prime Factors of 224:

224 = 2 × 2 × 2 × 2 × 2 × 7 = 25 × 7

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 224Comes out of the rootStays under the root
25222
71—7

3. Take the pairs out of the root.

√224 = √((2 × 2) × (2 × 2) × 2 × 7)

= 2 × 2 × √(2 × 7)

= 4√14

4. Check by squaring back.

4² × 14 = 16 × 14 = 224

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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 224 simplified? (The answer is: 4√14). Enter a whole number (e.g. '224') and hit the 'Simplify' button.
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Simplified Square Roots Table

NumberSimplest radical formDecimal value
209√209≈ 14.457
210√210≈ 14.491
211√211≈ 14.526
2122√53≈ 14.56
213√213≈ 14.595
214√214≈ 14.629
215√215≈ 14.663
2166√6≈ 14.697
217√217≈ 14.731
218√218≈ 14.765
219√219≈ 14.799
2202√55≈ 14.832
221√221≈ 14.866
222√222≈ 14.9
223√223≈ 14.933
2244√14≈ 14.967
2251515
226√226≈ 15.033
227√227≈ 15.067
2282√57≈ 15.1
229√229≈ 15.133
230√230≈ 15.166
231√231≈ 15.199
2322√58≈ 15.232
233√233≈ 15.264
2343√26≈ 15.297
235√235≈ 15.33
2362√59≈ 15.362
237√237≈ 15.395
238√238≈ 15.427

FAQ

What is the square root of 224 simplified?

√224 = 4√14

Is 224 a perfect square?

No. 224 = 16 × 14: its largest perfect square factor is 16 = 4², and the factor 14 stays under the radical, so √224 = 4√14.

What is √224 as a decimal?

√224 ≈ 14.967. The decimal never ends and never repeats, so it is always rounded; the exact value is 4√14.

What is the largest perfect square that divides 224?

16 = 4². Dividing gives 224 ÷ 16 = 14, and 14 has no square factor left, so √224 = √16 × √14 = 4√14.

See Also