What Is A Square Root?
A square root of a number $n$ is a value $r$ with $r^2 = n$. The square root of 120 is the number that, squared, gives 120.
No whole number does this: $10^2 = 100$ and $11^2 = 121$. So $\sqrt{120}$ sits between 10 and 11, very close to 11.
Where Does √120 Appear?
$\sqrt{120}$ turns up wherever a distance or a diagonal lands on 120 under the Pythagorean theorem, and as the side of a square whose area is 120 square units. Because $120 = 4 \times 30$, the value can be rewritten as $2\sqrt{30}$, which is how it usually appears in a geometry answer that must stay exact.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Simplest Radical | Rational or Irrational |
|---|---|---|---|
100 | 10 | 10 | Rational |
105 | 10.247 | $\sqrt{105}$ | Irrational |
115 | 10.724 | $\sqrt{115}$ | Irrational |
120 | 10.954 | $2\sqrt{30}$ | Irrational |
121 | 11 | 11 | Rational |
125 | 11.180 | $5\sqrt{5}$ | Irrational |
Is The Square Root Of 120 Rational Or Irrational?
$\sqrt{120}$ is irrational. After simplifying to $2\sqrt{30}$, the leftover $\sqrt{30}$ has no whole-number value, so the product is irrational.
The factorisation shows why. $$120 = 2^3 \times 3 \times 5$$
The primes 2, 3, and 5 cannot all be paired, so 120 is not a perfect square, and its root is an irrational number. Every non-perfect-square in the square root 1 to 30 list gives the same non-terminating decimal.
How Do You Find √120? (Prime Factorisation And Long Division)
Prime factorisation gives the exact form. Break 120 into primes, then pair what you can.
Step 1: Factor 120. $$120 = 2 \times 2 \times 2 \times 3 \times 5$$
Step 2: Group one pair of 2s. $$120 = 2^2 \times 30$$
Step 3: Pull the pair out of the radical. $$\sqrt{120} = \sqrt{2^2 \times 30} = 2\sqrt{30}$$
The leftover 30 has no repeated prime, so $2\sqrt{30}$ is fully reduced — the same simplifying radical expressions rule at work.
Long division gives the decimal, running like ordinary long division.
Step 1: Pair the digits. $$\overline{1}\ \overline{20}.\ \overline{00}$$
Step 2: The largest square at most 1 is 1. $$1^2 = 1$$
Step 3: Subtract and bring down 20. $$1 - 1 = 0 \rightarrow 20$$
Step 4: Double the quotient 1 to 2. No digit $d$ makes $(20 + d)\times d \le 20$ except 0, so the next quotient digit is 0. $$20 \times 0 = 0$$
Step 5: The integer part is 10, remainder 20. Continue with pairs of zeros. $$10.9,\ 10.95,\ 10.954$$
So $\sqrt{120} \approx 10.954$, matching $2\sqrt{30}$.
Examples Of √120
Example 1
Write $\sqrt{120}$ in simplest radical form.
Factor out the largest perfect square. $$120 = 4 \times 30$$ $$\sqrt{120} = \sqrt{4}\times\sqrt{30} = 2\sqrt{30}$$
The simplest radical form is $2\sqrt{30}$.
Example 2
A student simplifies $\sqrt{120}$ as $\sqrt{100}\times\sqrt{20} = 10 \times \sqrt{20}$. Why is this not simplest form?
Follow the wrong path. Choosing 100 as the factor looks tidy, but 100 is not a factor of 120, and $\sqrt{20}$ still hides a perfect square.
Test it. $$100 \times 20 = 2000 \neq 120$$
So the split is wrong from the start. Even fixing the arithmetic, $\sqrt{20} = 2\sqrt{5}$ can still be reduced.
The fix: factor out the largest perfect square that truly divides 120, which is 4. $$\sqrt{120} = 2\sqrt{30}$$
Example 3
Estimate $\sqrt{120}$ to the nearest tenth.
Bracket it with perfect squares. $$10^2 = 100$$ $$11^2 = 121$$
120 is almost 121, so the root is just below 11. $$\sqrt{120} \approx 11.0$$
Example 4
Confirm that $2\sqrt{30}$ equals $\sqrt{120}$ by squaring.
Square the simplified form. $$(2\sqrt{30})^2 = 2^2 \times (\sqrt{30})^2 = 4 \times 30 = 120$$
Since the square is 120, $2\sqrt{30} = \sqrt{120}$.
Example 5
A square garden has an area of 120 square feet. Give its exact and approximate side length.
The side is the square root of the area. $$s = \sqrt{120} = 2\sqrt{30}$$ $$s \approx 10.95 \text{ ft}$$
The exact side is $2\sqrt{30}$ feet, about 10.95 feet.
Common Mistakes
Mistake 1: Factoring out a non-factor
Where it slips in: Reaching for a big round perfect square like 100.
Don't do this: Writing $\sqrt{120} = \sqrt{100}\times\sqrt{20}$, since 100 does not divide 120.
The correct way: Use the largest perfect square that actually divides 120, which is 4, giving $2\sqrt{30}$.
Mistake 2: Stopping before fully reduced
Where it slips in: Splitting 120 as $\sqrt{4}\times\sqrt{30}$ but then re-splitting 30.
Don't do this: Trying to reduce $\sqrt{30}$ further.
The correct way: $30 = 2 \times 3 \times 5$ has no repeated prime, so $\sqrt{30}$ stays put and the answer is $2\sqrt{30}$, which is where learners who love to keep factoring overshoot.
Mistake 3: Turning √30 into 30
Where it slips in: Reading $2\sqrt{30}$ as $2 \times 30$.
Don't do this: Writing $2\sqrt{30} = 60$.
The correct way: $\sqrt{30} \approx 5.477$, so $2\sqrt{30} \approx 10.954$, not 60.
Conclusion
The square root of 120 is $2\sqrt{30}$, about 10.954, and irrational.
The perfect square 4 factors out of 120, leaving $\sqrt{30}$ inside.
Prime factorisation gives the exact form; long division gives the decimal.
Factor out the largest true perfect square — the trap in Example 2.
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