The lesson
This lesson teaches Roots & Irrational Numbers. Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.
Square roots and cube roots
The square root of 49 is 7 because 7 × 7 = 49. The cube root of 8 is 2 because 2 × 2 × 2 = 8.
When you study square roots and cube roots, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
Rational vs. irrational
A ratio compares two quantities with the same units. Order matters: a ratio of cats to dogs is not the same as dogs to cats unless the problem says they are equivalent.
Write ratios in three ways: with a colon (3:4), as a phrase (3 to 4), or as a fraction (3/4) when it fits the context.
When you study rational vs. irrational, slow down and write one example in your notebook without looking at the screen. That active step is what turns reading into learning.
- 1Rational numbers can be written as a fraction; their decimals end or repeat (like 0.5 or 0.333…).
- 2Irrational numbers cannot; their decimals go on forever without repeating (like π or √2).
A ratio compares two quantities with the same units. Order matters: a ratio of cats to dogs is not the same as dogs to cats unless the problem says they are equivalent.
Write ratios in three ways: with a colon (3:4), as a phrase (3 to 4), or as a fraction (3/4) when it fits the context.
Why this matters
Roots & Irrational Numbers shows up constantly in evaluate square and cube roots and tell rational from irrational numbers. It also connects to what you will see on homework, quizzes, and the next unit in this grade.
Teachers often move fast in class. This page is here so you can pause, re-read, and practice until the idea feels familiar, not just until you have memorized a rule for one day.
Common mistakes to avoid
Rushing to the answer without writing steps. Middle-school math rewards clear work, and you catch errors earlier when steps are visible.
Mixing up similar ideas from the same topic. If two terms feel alike, make a two-column note: what is the same, what is different, and one example of each.
Key ideas from this lesson
- Square roots and cube roots
- Rational vs. irrational
- Rational numbers can be written as a fraction; their decimals end or repeat (like 0.5 or 0.333…).
- Irrational numbers cannot; their decimals go on forever without repeating (like π or √2).
Video walkthrough
Introduction to Square Roots
What the radical symbol means and how to find square roots.
Watch on YouTubePractice
For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.
Exercise 1
Try it yourselfWhat is √81?
Step-by-step solution
- 1Ask: ? × ? = 81.
- 29 × 9 = 81.
- 3√81 = 9.
Exercise 2
Try it yourselfWhat is ∛27?
Step-by-step solution
- 1Cube root means x³ = 27.
- 23³ = 27.
- 3∛27 = 3.
Exercise 3
Try it yourselfBetween which two consecutive integers does √50 lie?
Step-by-step solution
- 17² = 49 and 8² = 64.
- 249 < 50 < 64, so 7 < √50 < 8.
- 3√50 is between 7 and 8.
Exercise 4
Try it yourselfClassify √16 as rational or irrational.
Step-by-step solution
- 1√16 = 4, a whole number.
- 24 = 4/1, so it is a ratio of integers.
- 3√16 is rational.
Exercise 5
Try it yourselfOrder from least to greatest: π, 3.14, √10, 3.
Step-by-step solution
- 1√10 ≈ 3.162; π ≈ 3.142; 3.14 = 3.140.
- 23 < 3.14 < π < √10.
- 3Order: 3, 3.14, π, √10.
Quick check
Answer all questions. Retake the quiz until you feel confident before moving on.
Roots & Irrational Numbers
Question 1 of 4
What is √64?