Square Root of 135 Simplified

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What is the square root of 135 simplified?

Answer

√135 = 3√15

135 = 9 × 15, and 9 = 3² is the largest perfect square that divides 135, so √135 = √9 × √15 = 3√15. Check: 3√15 ≈ 11.619, the same as √135 ≈ 11.619.

Simplifying √135 Step by Step

1. Prime factorization. — All Prime Factors of 135:

135 = 3 × 3 × 3 × 5 = 33 × 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 135Comes out of the rootStays under the root
3333
51—5

3. Take the pairs out of the root.

√135 = √((3 × 3) × 3 × 5)

= 3 × √(3 × 5)

= 3√15

4. Check by squaring back.

3² × 15 = 9 × 15 = 135

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Simplified Square Roots Table

NumberSimplest radical formDecimal value
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
1322√33≈ 11.489
133√133≈ 11.533
134√134≈ 11.576
1353√15≈ 11.619
1362√34≈ 11.662
137√137≈ 11.705
138√138≈ 11.747
139√139≈ 11.79
1402√35≈ 11.832
141√141≈ 11.874
142√142≈ 11.916
143√143≈ 11.958
1441212
145√145≈ 12.042
146√146≈ 12.083
1477√3≈ 12.124
1482√37≈ 12.166
149√149≈ 12.207

FAQ

What is the square root of 135 simplified?

√135 = 3√15

Is 135 a perfect square?

No. 135 = 9 × 15: its largest perfect square factor is 9 = 3², and the factor 15 stays under the radical, so √135 = 3√15.

What is √135 as a decimal?

√135 ≈ 11.619. The decimal never ends and never repeats, so it is always rounded; the exact value is 3√15.

What is the largest perfect square that divides 135?

9 = 3². Dividing gives 135 ÷ 9 = 15, and 15 has no square factor left, so √135 = √9 × √15 = 3√15.

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