Square Root of 176 Simplified

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What is the square root of 176 simplified?

Answer

√176 = 4√11

176 = 16 × 11, and 16 = 4² is the largest perfect square that divides 176, so √176 = √16 × √11 = 4√11. Check: 4√11 ≈ 13.266, the same as √176 ≈ 13.266.

Simplifying √176 Step by Step

1. Prime factorization. — All Prime Factors of 176:

176 = 2 × 2 × 2 × 2 × 11 = 24 × 11

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 176Comes out of the rootStays under the root
2422—
111—11

3. Take the pairs out of the root.

√176 = √((2 × 2) × (2 × 2) × 11)

= 2 × 2 × √11

= 4√11

4. Check by squaring back.

4² × 11 = 16 × 11 = 176

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Simplified Square Roots Table

NumberSimplest radical formDecimal value
161√161≈ 12.689
1629√2≈ 12.728
163√163≈ 12.767
1642√41≈ 12.806
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

FAQ

What is the square root of 176 simplified?

√176 = 4√11

Is 176 a perfect square?

No. 176 = 16 × 11: its largest perfect square factor is 16 = 4², and the factor 11 stays under the radical, so √176 = 4√11.

What is √176 as a decimal?

√176 ≈ 13.266. The decimal never ends and never repeats, so it is always rounded; the exact value is 4√11.

What is the largest perfect square that divides 176?

16 = 4². Dividing gives 176 ÷ 16 = 11, and 11 has no square factor left, so √176 = √16 × √11 = 4√11.

See Also