Unit 3: Roots and the Real Number System
Undoing a square, undoing a cube, and the strange numbers that show up when neither one comes out even.
5 topics · 20 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
Perfect Squares and Square Roots
- How it worksHow does a square root undo the job that squaring just did?
- Worked exampleFind the square root of 81 and the square root of 144.
- Going furtherWhat happens to a perfect square's square root every time you move to the next perfect square?
- Common mistakeWhy doesn't the square root of a number ever come out negative when you write the radical sign alone?
Your turn1 of 2
A square rug has a side length of 15 feet, so its area is 15 times 15 square feet. What is the side length, found by taking the square root of the area?Estimating Square Roots That Aren't Perfect
- How it worksHow do you squeeze a square root between two perfect squares to estimate it?
- Worked exampleFind the square root of 50 to the nearest tenth by testing numbers between 7 and 8.
- Common mistakeWhy doesn't rounding the square root of 50 to 7, just because 49 is close, give you an exact answer?
- Going furtherWhat happens to your estimate's accuracy as you test numbers with more decimal places?
Your turn1 of 2
A square garden plot covers 69 square metres. Estimate the length of one side, to the nearest tenth of a metre.Cube Roots and Perfect Cubes
- How it worksHow does a cube root undo raising a number to the 3rd power?
- Worked exampleFind the cube root of 27 and the cube root of negative 8.
- Going furtherWhat happens to the cube root of a number as the number itself grows from 1 to 1000?
- Common mistakeWhy can a cube root come out negative when a square root of a positive number never does?
Your turn1 of 2
A cube-shaped storage box has a volume of 7 times 7 times 7 cubic feet. What is the length of one side?Irrational Numbers
- How it worksHow does assuming the square root of 2 is a fraction lead to a contradiction that proves it isn't?
- Worked exampleShow that the square root of 9 is rational while the square root of 10 is irrational.
- Common mistakeWhy isn't a long, non-repeating decimal on a calculator screen just a fraction with a lot of digits?
- Going furtherWhat happens when you add a rational number to an irrational number?
Your turn1 of 2
Is the square root of 3 a rational number or an irrational number?The Real Number System
- How it worksHow do the categories of natural numbers, whole numbers, integers, rationals, and irrationals nest inside each other?
- Worked exampleFind the smallest category each of the numbers negative 3, 0.75, the square root of 16, and the square root of 5 belongs to.
- Common mistakeWhy isn't every square root automatically an irrational number?
- Ties togetherHow does the real number line you built with integers connect to every category in the real number system?
Your turn1 of 2
Which category does negative 14 belong to: natural numbers, whole numbers, integers, or irrational numbers?