Roots, Radicals, and Irrational Numbers

Learn about irrational numbers and their properties. Math 10-C Alberta mathematics curriculum.

Lesson 2.2 of Number Systems & Operations in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Understanding Roots and Radicals. Learn what roots are, how they reverse exponents, and the names of each part of a radical expression — plus the rule for negative radicands with even and odd indices.
• Exact vs. Approximate Values. Discover when a radical evaluates to a rational number (exact) versus an irrational one (approximate), and practice classifying radicals using perfect powers.
• Ordering Irrational Numbers on a Number Line. Compare and order radicals by converting them to decimal approximations, then place them correctly on a number line.

Inside the lesson — a free preview

What is a Root?

What is a Root?. A root is the reverse of an exponent. When you raise a number to a power, you multiply it by itself. A root asks the opposite question: what number, multiplied by itself a certain number of times, gives you this value?

Roots and Exponents — The Connection. Every root statement is just an exponent statement read backwards.

Since 3^2 = 9, we say 3 is a square root of 9, and write:

3 = √(9)

Since 3^3 = 27, we say 3 is a cube root of 27, and write:

3 = √(27)

Since 3^4 = 81, we say 3 is a fourth root of 81, and write:

3 = √(81)

The pattern continues for fifth roots, sixth roots, and beyond.

The Parts of a Radical

Vocabulary — Naming the Parts of √(a). Every radical expression √(a) has three named parts:

Radical sign — the x symbol itself.

Radicand — the number a underneath the radical sign. It is the number you are taking the root of.

Index — the small number n written above and to the left of the radical sign. It tells you which root you are finding (square root, cube root, fourth root, etc.).

Important: When no index is written, the index is 2 (a square root). So √(9) means √(9). You only need to write the index for cube roots and higher.

Writing Numbers as Different Roots

Example — Writing 5 as Square, Cube, and Fourth Roots. To write a number as a root, reverse the exponent process.

Step 1 — Square root:

5^2 = 25, so 5 = √(25)

Step 2 — Cube root:

5^3 = 125, so 5 = √(125)

Step 3 — Fourth root:

5^4 = 625, so 5 = √(625)

To write any number as a square root, square it and put the result under the radical. To write it as a cube root, cube it. And so on.

Worked example: Worked Example — Order from Least to Greatest

Order √(2),\ √(2),\ √(6),\ √(11),\ √(30) from Least to Greatest. Step 1 — Approximate each value.

√(2) ≈ 1.414

√(2) ≈ 1.260

√(6) ≈ 1.817

√(11) ≈ 3.317

√(30) ≈ 2.340

Step 2 — Order from least to greatest.

√(2) < √(2) < √(6) < √(30) < √(11)

1.260 < 1.414 < 1.817 < 2.340 < 3.317

Step 3 — Place on a number line.

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