The lesson
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.
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.
- 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).
Why this matters
Evaluate square and cube roots and tell rational from irrational numbers.
These ideas show up on homework, quizzes, and the next unit in this grade. Getting comfortable here saves time later when the problems stack more steps together.
Common mistakes to avoid
Reversing the order named in the problem. Write the ratio in the same order the words give you.
Treating a part-to-part ratio like a part-to-whole fraction without checking what the whole is.
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 walkthroughs
Introduction to Square Roots
What the radical symbol means and how to find square roots.
Watch on YouTube
Approximating Square Roots
Estimate irrational square roots to the nearest hundredth.
Watch on YouTube
Sums and Products of Irrational Numbers
Explore when roots and pi produce rational or irrational results.
Watch on YouTube
The Pythagorean Theorem
Use square roots to find a missing side in a right triangle.
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?