Squares and Square Roots: Basics of Working with Roots

The square root is the inverse operation of squaring. While a square tells you how large an area is for a given side length, a square root tells you the length of the side of a square with a given area.

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Table of Contents

• What is the square root
• Square root symbol
• Relationship between powers and roots
• Examples of square roots
• Square root – practical examples
• What is the cube root?
• Common mistakes
• Rules for roots
• Rational and irrational numbers
• Frequently Asked Questions

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What is the square root

The square root of a number $a$ is a number $b$ such that:

We denote it as:

Example: $\sqrt{16}=4$, because $4^2=16$

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Square root symbol

The symbol ​ is called the radical (or root symbol). The number under the radical is called the radicand.

```

┌─┐

│16│ ← radicand (number under the radical)

│ 4 │ ← radical symbol

└─┘

```

Small version vs. large version

• $\sqrt{16}$ = square root of 16 (most common)
• $\sqrt[3]{8}$ = cube root of 8
• $\sqrt[n]{a}$ = $n$-th root of $a$

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Relationship between powers and roots

The square and the square root are inverse operations:

Relationship between powers and roots:

• Square: $4^2=16$
• Square root: $\sqrt{16}=4$

We can write:

(the absolute value, because $\sqrt{a^2}$ is always non-negative)

But be careful:

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Examples of square roots

Numbers with "nice" square roots

Numbers with "nice" square roots:

• $\sqrt{1}=1$
• $\sqrt{4}=2$
• $\sqrt{9}=3$
• $\sqrt{16}=4$
• $\sqrt{25}=5$
• $\sqrt{36}=6$
• $\sqrt{49}=7$
• $\sqrt{64}=8$
• $\sqrt{81}=9$
• $\sqrt{100}=10$
• $\sqrt{121}=11$
• $\sqrt{144}=12$
• $\sqrt{169}=13$

Numbers without "nice" square roots

Not all numbers have "nice" square roots:

• $\sqrt{2}\approx 1.414$ (irrational)
• $\sqrt{3}\approx 1.732$ (irrational)
• $\sqrt{5}\approx 2.236$ (irrational)
• $\sqrt{7}\approx 2.646$ (irrational)

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Square root practical examples

Geometry – finding side length

If a square has an area of $64$ cm², what is the length of its side?

Physics – period of a pendulum

The period of a pendulum is:

where $L$ is the length and $g$ is gravitational acceleration.

Pythagorean theorem

In a right triangle:

So to find $c$:

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What is the cube root

While the square root "undoes" squaring, the cube root "undoes" cubing:

Cube root:

Examples

• $\sqrt[3]{8}=2$ because $2^3=8$
• $\sqrt[3]{27}=3$ because $3^3=27$
• $\sqrt[3]{125}=5$ because $5^3=125$
• $\sqrt[3]{-8}=-2$ because $(-2{)}^3=-8$

Note on negative numbers

Unlike square roots, cube roots of negative numbers ARE defined:

$\sqrt[3]{-8}=-2$ because $(-2{)}^3=-8$

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Common mistakes

Mistake 1: Confusing $\sqrt{a^2}$ with $a$

For any $a$: $\sqrt{a^2}=|a|$ (always non-negative)

Mistake 2: Thinking $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$ always works

Actually, this IS true:

But only when $a\ge 0$ and $b\ge 0$.

Mistake 3: Forgetting that $\sqrt{4}=\pm 2$ in equations

When solving $x^2=4$, we get $x=\pm 2$.

But $\sqrt{4}=2$ (only the principal, positive root).

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Rules for roots

Product rule

Example: $\sqrt{16\cdot 9}=\sqrt{144}=12$

Also: $\sqrt{16}\cdot \sqrt{9}=4\cdot 3=12$ ✓

Quotient rule

Example: $\sqrt{\frac{100}{4}}=\sqrt{25}=5$

Also: $\frac{\sqrt{100}}{\sqrt{4}}=\frac{10}{2}=5$ ✓

Power rule

Example: $\sqrt[3]{2^6}=2^{\frac{6}{3}}=2^2=4$

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Rational and irrational numbers

Rational numbers

Numbers that can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers ($q\ne 0$).

Examples: $2$, $-3$, $\frac{3}{4}$, $\mathrm{0.666...}$, $0.5$

Irrational numbers

Numbers that CANNOT be written as a fraction of integers.

Examples: $\sqrt{2}$, $\sqrt{3}$, $\pi$, $e$

Important facts

• $\sqrt{1}=1$ (rational)
• $\sqrt{2}\approx 1.414$ (irrational) – this was the first number proven to be irrational!
• $\sqrt{3}\approx 1.732$ (irrational)
• $\sqrt{4}=2$ (rational)
• $\sqrt{5}\approx 2.236$ (irrational)

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Frequently Asked Questions

What is the difference between $\sqrt{9}$ and $\pm \sqrt{9}$?

• $\sqrt{9}=3$ (the principal square root, always positive)
• $\pm \sqrt{9}=\pm 3$ (both solutions to the equation $x^2=9$)

Can we take the square root of a negative number?

In real numbers, no! $\sqrt{-4}$ is undefined.

But in complex numbers: $\sqrt{-4}=2i$

Is $\sqrt{a^2}=a$ always true?

No! $\sqrt{a^2}=|a|$

For example: $\sqrt{(-3{)}^2}=\sqrt{9}=3=|-3|$

Why do we use square roots?

Square roots are used in geometry (Pythagorean theorem), physics (period of pendulum, kinetic energy), statistics (standard deviation), and many other fields.

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Summary

• Operation: Square → Notation: $a^2$ → Example: $4^2=16$
• Operation: Square root → Notation: $\sqrt{a}$ → Example: $\sqrt{16}=4$
• Operation: Cube → Notation: $a^3$ → Example: $2^3=8$
• Operation: Cube root → Notation: $\sqrt[3]{a}$ → Example: $\sqrt[3]{8}=2$

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Practice

Test yourself with interactive exercises:

• 🔢 Basic powers
• 📊 Comparing powers
• 🧮 Calculating with powers