Square Root of 25 — Value, Method, and Examples
Square Root of 25 — Value, Method, and Examples
TL;DR
The square root of 25 is exactly 5, because 5×5=25 — making 25 a perfect square with a whole-number root. This article gives the value, three ways to find it, where 25 turns up, and why the radical answer is 5 rather than ±5.
The Square Root of 25 Is 5
The square root of 25 is 5. It is a perfect square: 5×5=25, so 25=5 with no decimal tail and no rounding.
This is one of the first square roots worth knowing by heart — it anchors the 3–4–5 triangle, the most-used right triangle in school geometry.
Quick Answer
Quick Answer:
Result: 25=5
Notation: radical form 25; exponent form 25^{1/2}.
Method shown: prime factorisation, with cross-checks by repeated subtraction and long division.
Rational or irrational: rational — 5 can be written as \tfrac{5}{1}.
Exact form: 5 (an integer; no radical remains).
Quick Reference Table — Square Roots of Nearby Perfect Squares
| n | n√n | Perfect square? |
|---|---|---|
| 1 | 1 | yes |
| 4 | 2 | yes |
| 9 | 3 | yes |
| 16 | 4 | yes |
| 25 | 5 | yes |
| 36 | 6 | yes |
| 49 | 7 | yes |
| 64 | 8 | yes |
| 81 | 9 | yes |
| 100 | 10 | yes |
Every value in this table is a whole number — these ten are the perfect squares from 1 to 100, and knowing them on sight makes most square-root work fast.
Where the square root of 25 appears
The 25 result sits at the center of the 3–4–5 right triangle: legs of 3 and 4 give a hypotenuse of \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5. That single triple shows up in the Pythagorean theorem, in carpentry for squaring a corner, and in the distance between the points (0,0) and (3,4). It also appears whenever a quadratic has a discriminant of 25, since 25 keeps the roots rational.
What "square root of 25" Means
A square root of a number n is a value x for which x^2 = n. For 25, the value is 5, since 5^2 = 25.
The radical symbol √ asks for the principal (non-negative) root, so 25=5. The equation x^2=25 has two solutions, x=5 and x=−5, but the symbol 25 alone names only the positive one.
How To Find The Square Root of 25
Method 1 — Prime factorisation
Break 25 into prime factors, then pair them.
25=5×5
One factor leaves each pair, so 25=5. This is the cleanest method for any perfect square.
Final answer: 25=5.
Method 2 — Repeated subtraction of odd numbers
Subtract successive odd numbers from 25 and count the steps until you reach 0.
25−1=24, 24−3=21, 21−5=16, 16−7=9, 9−9=0.
Five subtractions reached zero, so 25=5. (This works because n^2 is the sum of the first n odd numbers.)
Method 3 — Long division
Pair the digits: 25. The largest integer whose square is at most 25 is 5, and 5^2 = 25 exactly. Subtracting leaves a remainder of 0, so the root terminates.
Final answer: 25=5.
Examples of Square Root of 25
Example 1
Evaluate 25 directly.
5^2 = 25, so 25=5. A clean integer — no approximation needed.
Example 2
A student is asked: does x^2=25 mean x=25?
Wrong attempt. Writing x=25 as the only answer drops a solution. Check: (−5)^2=25 as well, so −5 also satisfies the equation.
Correct. The equation x^2=25 has two roots, x=±5. The symbol 25 names only the principal root, 5. The two ideas are different — solving an equation is not the same as evaluating a radical.
Example 3
Simplify 3√25.
25=5, so 3×5=15.
Example 4
Find the hypotenuse of a right triangle with legs 3 and 4.
c=\sqrt{3^2 + 4^2} = \sqrt{25} = 5.
Example 5
Solve 25x^2 for x>0.
25x^2=5x. The square root distributes over a product (unlike over a sum).
Where Students Trip Up on the Square Root of 25
Mistake 1: Writing √25 as ±5
Where it slips in: A student confuses solving x^2=25 with evaluating the radical 25.
Don't do this: 25=±5.
The correct way: 25=5. The radical returns only the principal (positive) root; the ± belongs to equation-solving, not to the symbol itself.
Mistake 2: Confusing the square root with halving
Where it slips in: Treating 25 as 25÷2.
Don't do this: 25=12.5.
The correct way: 25 asks "what times itself gives 25?" — the answer is 5, not half of 25.
Mistake 3: Calling √25 irrational
Where it slips in: Lumping every square root in with other non-terminating square roots.
Don't do this: Assuming 25 never terminates.
The correct way: 25 is a perfect square, so 25=5 is a whole number — clearly rational.
Conclusion
The square root of 25 is 5, an exact whole number, because 25 is a perfect square (5×5=25).
25 names only the principal root, 5; the equation x^2=25 is what gives ±5.
Prime factorisation is the fastest route: 25=5×5, so 25=5.
Because 5 is an integer, 25 is rational, not irrational.
25=5 is the heart of the 3–4–5 right triangle.
Frequently Asked Questions
Is the square root of 25 rational or irrational?
Rational. 25=5, and 5 can be written as the fraction \tfrac{5}{1}.
What is the square root of 25 in radical form?
It doesn't stay in radical form — 25 simplifies to the integer 5 because 25 is a perfect square.
What is −25?
There is no real square root of a negative number.