Square Root of 208 Simplified

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What is the square root of 208 simplified?

Answer

√208 = 4√13

208 = 16 × 13, and 16 = 4² is the largest perfect square that divides 208, so √208 = √16 × √13 = 4√13. Check: 4√13 ≈ 14.422, the same as √208 ≈ 14.422.

Simplifying √208 Step by Step

1. Prime factorization. — All Prime Factors of 208:

208 = 2 × 2 × 2 × 2 × 13 = 24 × 13

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 208Comes out of the rootStays under the root
2422—
131—13

3. Take the pairs out of the root.

√208 = √((2 × 2) × (2 × 2) × 13)

= 2 × 2 × √13

= 4√13

4. Check by squaring back.

4² × 13 = 16 × 13 = 208

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Simplified Square Roots Table

NumberSimplest radical formDecimal value
193√193≈ 13.892
194√194≈ 13.928
195√195≈ 13.964
1961414
197√197≈ 14.036
1983√22≈ 14.071
199√199≈ 14.107
20010√2≈ 14.142
201√201≈ 14.177
202√202≈ 14.213
203√203≈ 14.248
2042√51≈ 14.283
205√205≈ 14.318
206√206≈ 14.353
2073√23≈ 14.387
2084√13≈ 14.422
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

FAQ

What is the square root of 208 simplified?

√208 = 4√13

Is 208 a perfect square?

No. 208 = 16 × 13: its largest perfect square factor is 16 = 4², and the factor 13 stays under the radical, so √208 = 4√13.

What is √208 as a decimal?

√208 ≈ 14.422. The decimal never ends and never repeats, so it is always rounded; the exact value is 4√13.

What is the largest perfect square that divides 208?

16 = 4². Dividing gives 208 ÷ 16 = 13, and 13 has no square factor left, so √208 = √16 × √13 = 4√13.

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