Square Roots · Evaluate & estimate · 2.5

Compare & order roots.

To compare roots with whole numbers, compare their squares (or estimate each). Since 3 = √9, ordering √8, 3, √15 is ordering √8, √9, √15 → √8, 3, √15.

01 / THE IDEA

See it work.

Watch the method

Compare & order roots

3 = √9, so √8 < 3 < √15.

1 / 2
8,  3,  15\sqrt{8},\; 3,\; \sqrt{15}
1

Estimate each value

Find a decimal value (or nearest squares) for each.

8,  3,  15\sqrt{8},\; 3,\; \sqrt{15}
2

Order from least to greatest

Line them up by size.

8<3<15\sqrt{8} < 3 < \sqrt{15}
— / WHY IT WORKS

The idea underneath.

Same form to compare

Write whole numbers as roots (3 = √9) or estimate each value, then order from least to greatest.

Why it works

√8 < √9 < √15.

03 / WORKED EXAMPLES

From easy to tricky.

EX 1easy

Order from least to greatest: √5, 2, √24

1

Estimate each value

Find a decimal value (or nearest squares) for each. √5,\; 2,\; √24

2

Order from least to greatest

Line them up by size. 2 < √5 < √24

EX 2medium

Order from least to greatest: √2, 4, √18

1

Estimate each value

Find a decimal value (or nearest squares) for each. √2,\; 4,\; √18

2

Order from least to greatest

Line them up by size. √2 < 4 < √18

EX 3hard

Order from least to greatest: √18, 3, √3

1

Estimate each value

Find a decimal value (or nearest squares) for each. √18,\; 3,\; √3

2

Order from least to greatest

Line them up by size. √3 < 3 < √18

04 / COMMON TRAPS

Where students slip.

Trap

Comparing radicands and roots directly.

Fix

Put everything in the same form first.

Trap

Forgetting 3 = √9.

Fix

A whole number n equals √(n²).

05 / QUIZ

Test yourself.

The Test

12 problems across three tiers. Auto-graded on submit. Hints and a full walkthrough on every question. Your work is saved on this device — sign in to keep it across devices.

0/12Answered
How to answerType just the value. Formatting won't trip you up — we ignore spaces, capital letters, commas, and subscript style, so t₁ = 5, d = 3 and t1=5,d=3 both count. For a list of numbers, separate them with commas (e.g. 3, 10, 17).
Core

Core (4)

C1

Order from least to greatest: √24, 2, √10

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
C2

Order from least to greatest: √10, 4, √3

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
C3

Order from least to greatest: √3, 3, √32

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
C4

Order from least to greatest: √8, 3, √3

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
Apply

Apply (4)

A1

Order from least to greatest: √8, 2, √24

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
A2

Order from least to greatest: √12, 2, √24

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
A3

Order from least to greatest: √6, 5, √32

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
A4

Order from least to greatest: √20, 3, √8

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
Extension

Extension (4)

E1

Order from least to greatest: √5, 3, √24

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
E2

Order from least to greatest: √24, 5, √3

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
E3

Order from least to greatest: √20, 2, √27

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.
E4

Order from least to greatest: √20, 5, √5

Need a hint?
  • Write each whole number as a root (n = √n²).
  • Then order the radicands.