Square Root of 100 — Value, Method & Examples

Square Root of 100 — Value, Method & Examples

TL;DR

The square root of 100 is exactly 10, because 10×10=100 — so 100 is a perfect square with a clean whole-number root. This article gives the value, three ways to compute it, where ( \sqrt{100} ) shows up, and the slip-ups students make most often.

The Square Root of 100 is 10

The square root of 100 is 10. It is a perfect square: 10×10=100, so ( \sqrt{100} = 10 ) — no decimal, no rounding.

It is also the round-number anchor of the square-root table from 1 to 100: ( \sqrt{100} ) closes the run of perfect squares that starts at 1.

Quick Answer:

Result: ( \sqrt{100} = 10 ) Notation: radical form ( \sqrt{100} ); exponent form ( 100^{1/2} ). Method shown: prime factorisation, with cross-checks by long division and repeated subtraction. Rational or irrational: rational — ( 10 = \frac{10}{1} ). Exact form: 10 (an integer; the radical fully resolves).

Quick Reference Table — Square Roots of Nearby Perfect Squares

n ( \sqrt{n} ) Perfect square?
49 7 yes
64 8 yes
81 9 yes
100 10 yes
121 11 yes
144 12 yes
169 13 yes
196 14 yes
225 15 yes
256 16 yes

These are the perfect squares bracketing 100, with whole-number roots on either side — useful for estimating any nearby non-perfect square at a glance.

Where The Square Root of 100 Appears

The result ( \sqrt{100} = 10 ) is everywhere the decimal system is, because 100=10² is the base of our place value — hundreds, percentages, and centi-units all rest on it. Geometrically, a square of area 100 square units has a side of ( \sqrt{100} = 10 ) units. It also turns up in the Pythagorean theorem: a 6–8–10 right triangle has a hypotenuse of 10, which is just the 3–4–5 triangle scaled by two.

What "square root of 100" Means

A square root of a number ( n ) is a value ( x ) for which ( x^2 = n ). For 100, that value is 10, since ( 10^2 = 100 ).

The radical symbol ( \sqrt{;}; ) returns the principal (non-negative) root, so ( \sqrt{100} = 10 ). The equation ( x^2 = 100 ) has two solutions, 10 and −10, but the symbol ( \sqrt{100} ) alone names only the positive one.

How To Find The Square Root of 100

Method 1 — Prime factorisation

Factor 100 into primes, pair them, and take one from each pair.

( 100 = 2 \times 2 \times 5 \times 5 = (2 \times 2)(5 \times 5) )

One 2 and one 5 leave their pairs: ( \sqrt{100} = 2 \times 5 = 10 ).

Final answer: ( \sqrt{100} = 10 ).

Method 2 — Long division

Pair the digits from the right: ( 1,00 ). The largest integer whose square is at most 1 is 1, so the first quotient digit is 1, with remainder 0. Bring down 00 to get a dividend of 00; double the quotient (1 → 2) and find ( d ) with ( (20+d) \cdot d \le 0 ), which gives ( d=0 ). The quotient is 10 with remainder 0, so ( \sqrt{100} = 10 ) exactly.

Final answer: ( \sqrt{100} = 10 ).

Method 3 — Repeated subtraction of odd numbers

Subtract successive odd numbers and count the steps to reach 0: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19.

That is 10 odd numbers, and 1+3+⋯+19=100, so ( \sqrt{100} = 10 ).

Examples of Square Root of 100

Example 1

Evaluate ( \sqrt{100} ) directly.

( 10^2 = 100 ), so ( \sqrt{100} = 10 ). A whole number — no estimation needed.

Example 2

A student is asked to simplify ( 100 + \sqrt{100} + \sqrt{100} ).

Wrong attempt. Reading it as "the root of 100 + 100 + 100," the student writes ( 200 \approx 14.14 ).

Correct. The two radicals are separate terms: ( 100 + \sqrt{100} + \sqrt{100} = 10 + 10 = 20 ). Square roots are evaluated before adding — the expression is ( 10 + 10 ), not ( 200 ).

Example 3

Simplify ( 100x^2 ) for x>0.

( \sqrt{100x^2} = \sqrt{100} \cdot \sqrt{x^2} = 10x ). The root distributes over a product.

Example 4

Find the side of a square whose area is 100 cm².

( \text{side} = \sqrt{100} = 10 ext{ cm} ). Area = side², so the side is the square root of the area.

Example 5

A right triangle has legs 6 and 8. Find the hypotenuse.

( c = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 ). The 6–8–10 triangle.

Common Mistakes With the Square Root of 100

Mistake 1: Confusing ( \sqrt{100} ) with 100 ÷ 2

Where it slips in: A learner mistakenly reaches for division instead of asking what value squares to 100.

Don't do this: ( \sqrt{100} = 50 ).

The correct way: ( \sqrt{100} ) asks "what number times itself gives 100?" — that's 10.

Mistake 2: Writing ( \sqrt{100} ) as ±10

Where it slips in: Confusing the radical with the solutions of ( x^2=100 ).

Don't do this: ( \sqrt{100} = ±10 ).

The correct way: ( \sqrt{100} = 10 ). The radical returns only the principal root.

Mistake 3: Splitting the root over a sum

Where it slips in: Treating ( 36+64 ) as ( \sqrt{36} + \sqrt{64} ).

Don't do this: ( 36+64=6+8=14 ).

The correct way: ( 36+64=100=\sqrt{100} = 10 ). Square roots do not distribute over addition.

Conclusion