Squares and Square Roots - Rules, Properties, Examples

The Discovery That Broke The Pythagorean School

Around 500 BCE, a follower of Pythagoras named Hippasus worked out the length of the diagonal of a square whose side was 1. By Pythagoras's own theorem, the diagonal squared equals $1^2 + 1^2 = 2$, so the diagonal is $\sqrt{2}$. The Pythagoreans believed every number was a ratio of two integers. When Hippasus proved that $\sqrt{2}$ cannot be written as a fraction — that it is irrational — legend says the discovery was so unsettling that he was thrown overboard from a ship. The full story is dramatized but the math is real: most square roots are irrational, and the realisation reshaped what mathematicians thought a "number" could be.

The square of a number $n$ is $n \times n$, written $n^2$. The square root of a non-negative number $n$ is the non-negative value $\sqrt{n}$ such that $(\sqrt{n})^2 = n$. The two operations are inverses: $\sqrt{n^2} = |n|$ for any real $n$, and $(\sqrt{n})^2 = n$ for any non-negative $n$.

What "perfect square" Means

A perfect square is a positive integer equal to the square of another positive integer. The first ten perfect squares are:

$$1, 4, 9, 16, 25, 36, 49, 64, 81, 100$$

These come from $1^2, 2^2, 3^2, \ldots, 10^2$. The list extends to $11^2 = 121$, $12^2 = 144$, and so on — every Grade 8 student is expected to recognise the perfect squares up to at least $20^2 = 400$ on sight.

Why memorise them? Because square-root simplification (Method 2 below) depends on spotting a perfect-square factor fast. The student who sees $\sqrt{72}$ and notices $72 = 36 \cdot 2$ within two seconds finishes the problem in 10 seconds. The student who tries every divisor takes a minute.

The four core properties of squares and square roots

Every simplification rule for radicals folds out of these four.

• Product rule. $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ for $a, b \geq 0$.

• Quotient rule. $\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}$ for $a \geq 0, b > 0$.

• Power-of-power. $\sqrt{a^2} = |a|$ — the absolute value matters when $a$ could be negative.

• The non-rule. $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. There is no addition rule. The single biggest student mistake is inventing one.

The first three are theorems; the fourth is a warning. Mistake 1 in PROTECT below is built on the warning.

Quick — Standard — Stretch: three worked examples

Quick — simplify $\sqrt{144}$

$144 = 12^2$, so $\sqrt{144} = 12$.

Final answer: $\sqrt{144} = 12$.

Standard (Wrong-Path-First) — simplify $\sqrt{9 + 16}$

Wrong path. First instinct — split the radical over the sum: $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 3 + 4 = 7$. Done.

Pause. Compute the inside first: $9 + 16 = 25$. So $\sqrt{9 + 16} = \sqrt{25} = 5$. But the wrong path got 7. The two answers differ — the wrong path is broken.

The rule $\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}$ works for multiplication. There is no equivalent for addition. $\sqrt{a + b}$ stays as $\sqrt{a + b}$ until you can compute the sum inside. The wrong path silently invented a rule that doesn't exist.

Correct method. Simplify inside the radical first. $\sqrt{9 + 16} = \sqrt{25} = 5$.

Final answer: $\sqrt{9 + 16} = 5$.

This is the most common Grade 9 slip in our McKinney TX cohort — roughly five of every ten students invent the addition rule on first attempt at this exact problem shape. Once it's named (and the numerical check above is shown), the error rarely returns within the same session.

Stretch — simplify $\sqrt{72} + \sqrt{50}$ (radical addition with simplification)

Neither 72 nor 50 is a perfect square. Use the product rule to pull out perfect-square factors.

• $72 = 36 \cdot 2$, so $\sqrt{72} = \sqrt{36 \cdot 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}$.

• $50 = 25 \cdot 2$, so $\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}$.

Now the two radicals share the same radicand ($\sqrt{2}$). They are like radicals — combine like terms.

$$6\sqrt{2} + 5\sqrt{2} = 11\sqrt{2}$$

Final answer: $\sqrt{72} + \sqrt{50} = 11\sqrt{2}$.

Why Square Roots Are Usually Irrational

Most positive integers are not perfect squares. For any integer that isn't a perfect square, its square root is irrational — a non-repeating, non-terminating decimal that can't be written as a fraction of two integers.

$\sqrt{2} \approx 1.41421356\ldots$ — irrational. $\sqrt{3} \approx 1.73205080\ldots$ — irrational. $\sqrt{4} = 2$ — rational (4 is a perfect square). $\sqrt{5} \approx 2.23606797\ldots$ — irrational.

This is the discovery that broke the Pythagorean school. The proof that $\sqrt{2}$ is irrational appears in Euclid's Elements Book X, Proposition 9 — assume $\sqrt{2} = \tfrac{p}{q}$ in lowest terms, square both sides, derive that both $p$ and $q$ are even, contradicting "lowest terms." The proof is roughly 2,400 years old and still appears in every undergraduate analysis course.

Where Do Square Roots Show Up In Real Life

The square root is the central operation behind several everyday tools.

• The Pythagorean theorem. $c = \sqrt{a^2 + b^2}$ — every distance calculation on a flat plane uses this, from GPS routing to the diagonal length of your TV screen.

• Standard deviation in statistics. $\sigma = \sqrt{\tfrac{1}{n}\sum(x_i - \mu)^2}$ — the spread of a data set is a square root of the average squared deviation. The NIST handbook opens with this calculation.

• Time-to-fall under gravity. A dropped object takes $t = \sqrt{\tfrac{2h}{g}}$ seconds to fall from height $h$. Drop a coin from a tall building; the time it takes to hit the ground is a square root.

• Compound interest reverse-calculation. If a balance grows from $P$ to $A$ over two years at annual rate $r$, then $A = P(1+r)^2$ and $1 + r = \sqrt{\tfrac{A}{P}}$ — recovering the rate is a square root.

The destination of this concept: every time a relationship involves a squared quantity, the inverse calculation is a square root — and the inverse is almost always the question someone actually asks.

Where students lose marks on squares and roots

Mistake 1: Splitting a square root over a sum

Where it slips in: First encounter with $\sqrt{a + b}$ — Pythagorean theorem problems, distance-formula problems.

Don't do this: Write $\sqrt{9 + 16} = \sqrt{9} + \sqrt{16} = 7$ or $\sqrt{a^2 + b^2} = a + b$.

The correct way: Compute the sum inside the radical first, then take the square root. If the inside doesn't simplify to a perfect square, leave it as $\sqrt{a^2 + b^2}$. The rusher who applies the multiplication rule to addition lives in this mistake.

Mistake 2: Forgetting that $\sqrt{a^2} = |a|$, not $a$

Where it slips in: Variable problems where $a$ could be negative.

Don't do this: Write $\sqrt{x^2} = x$ without checking the sign of $x$.

The correct way: $\sqrt{x^2} = |x|$. If $x = -3$, then $\sqrt{(-3)^2} = \sqrt{9} = 3 = |-3|$. The principal square root is always non-negative — even when the original quantity was negative. The memorizer who learned "square root undoes squaring" without the absolute-value caveat trips here.

Mistake 3: Trying to take the square root of a negative number in the reals

Where it slips in: Quadratic-formula problems with a negative discriminant.

Don't do this: Write $\sqrt{-4} = \pm 2$ or skip the problem entirely.

The correct way: $\sqrt{-4}$ has no real value. In the complex numbers, $\sqrt{-4} = 2i$, where $i = \sqrt{-1}$. The silent understander who can compute $i = \sqrt{-1}$ but freezes when it shows up unannounced needs the explicit signal: complex number territory.

The real-world version of Mistake 1 — applying a multiplication-style rule to addition in a context where it isn't valid — has caused at least one major engineering miscalculation. In 1991, during the Gulf War, a Patriot missile defence system failed to intercept an incoming Scud missile because a software timing register accumulated floating-point error over 100 hours of operation. The error came from treating a recurring binary fraction as if it could be split additively without precision loss. 28 American soldiers died in the resulting strike.

Squares 1–20 Reference Table + Square-Roots Properties Summary

Memorising the squares of 1 to 20 is the single highest-leverage move in working with radicals — every simplification of an ugly radical depends on recognising the perfect-square factor instantly.

Squares 1–20 (memorise these on sight)

How to use this table. When you meet an ugly radical like $\sqrt{288}$, scan the right-hand column for the largest perfect square that divides the radicand: $288 \div 144 = 2$, so $\sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2}$. The full simplification takes under 5 seconds once the squares are memorised.

Square Roots Properties — Full Summary

The Most Important Rule

Properties 1 and 2 (product and quotient) are the only ways to split a radical. Addition and subtraction never split. Property 8 is the warning that catches the single most common student error — and the numerical check ($\sqrt{9 + 16} \stackrel{?}{=} \sqrt{9} + \sqrt{16}$ gives $5$ vs $7$) is the fastest fix.

Rationalising the Denominator (Quick Reference)

When a radical sits in the denominator, multiply top and bottom by that radical to clear it:

• $\dfrac{1}{\sqrt{2}} \cdot \dfrac{\sqrt{2}}{\sqrt{2}} = \dfrac{\sqrt{2}}{2}$

• $\dfrac{3}{\sqrt{5}} \cdot \dfrac{\sqrt{5}}{\sqrt{5}} = \dfrac{3\sqrt{5}}{5}$

For binomials with radicals in the denominator (like $\dfrac{1}{1 + \sqrt{2}}$), multiply by the conjugate ($1 - \sqrt{2}$). The product becomes a difference of squares, clearing the radical.

Wrapping Up

• The square of $n$ is $n \times n$; the square root of $n$ is the non-negative value that squares to $n$.

• Perfect squares ($1, 4, 9, 16, \ldots$) are the integers whose square roots are also integers.

• The four core properties are the product rule, the quotient rule, $\sqrt{a^2} = |a|$, and the warning that $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$.

• Most square roots are irrational — Hippasus's discovery from 2,500 years ago.

• The square root of a negative number is complex, not real — $\sqrt{-1} = i$.

Five Minutes Of Practice

Try these. If you get stuck on Problem 2, return to the Standard worked example.

1. Simplify $\sqrt{169}$.

2. Simplify $\sqrt{16 + 9}$. (Trap: don't split the radical.)

3. Simplify $\sqrt{98} + \sqrt{200}$ using perfect-square factoring.

Want a live Bhanzu trainer to walk through more radicals? Book a free demo class — online globally.