Square Root of 117 Simplified

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What is the square root of 117 simplified?

Answer

√117 = 3√13

117 = 9 × 13, and 9 = 3² is the largest perfect square that divides 117, so √117 = √9 × √13 = 3√13. Check: 3√13 ≈ 10.817, the same as √117 ≈ 10.817.

Simplifying √117 Step by Step

1. Prime factorization. — All Prime Factors of 117:

117 = 3 × 3 × 13 = 32 × 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 117Comes out of the rootStays under the root
323—
131—13

3. Take the pairs out of the root.

√117 = √((3 × 3) × 13)

= 3 × √13

= 3√13

4. Check by squaring back.

3² × 13 = 9 × 13 = 117

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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 117 simplified? (The answer is: 3√13). Enter a whole number (e.g. '117') and hit the 'Simplify' button.
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Simplified Square Roots Table

NumberSimplest radical formDecimal value
102√102≈ 10.1
103√103≈ 10.149
1042√26≈ 10.198
105√105≈ 10.247
106√106≈ 10.296
107√107≈ 10.344
1086√3≈ 10.392
109√109≈ 10.44
110√110≈ 10.488
111√111≈ 10.536
1124√7≈ 10.583
113√113≈ 10.63
114√114≈ 10.677
115√115≈ 10.724
1162√29≈ 10.77
1173√13≈ 10.817
118√118≈ 10.863
119√119≈ 10.909
1202√30≈ 10.954
1211111
122√122≈ 11.045
123√123≈ 11.091
1242√31≈ 11.136
1255√5≈ 11.18
1263√14≈ 11.225
127√127≈ 11.269
1288√2≈ 11.314
129√129≈ 11.358
130√130≈ 11.402
131√131≈ 11.446

FAQ

What is the square root of 117 simplified?

√117 = 3√13

Is 117 a perfect square?

No. 117 = 9 × 13: its largest perfect square factor is 9 = 3², and the factor 13 stays under the radical, so √117 = 3√13.

What is √117 as a decimal?

√117 ≈ 10.817. The decimal never ends and never repeats, so it is always rounded; the exact value is 3√13.

What is the largest perfect square that divides 117?

9 = 3². Dividing gives 117 ÷ 9 = 13, and 13 has no square factor left, so √117 = √9 × √13 = 3√13.

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