Common Core Standards: Math
2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
Squares and cubes. They're not just for geometry anymore.
Students should already know that squaring a number means multiplying it by itself. So to square 3, we'd multiply 3 × 3, which we denote as 32. Likewise, cubing a number means multiplying it by itself twice. To cube 3, we'd multiply 3 × 3, and multiply that answer by 3 again. Essentially, what we'd have is 3 × 3 × 3, or 33.
Hopefully within the course of their (relatively short) lifetimes, your students have observed that that math is pretty logical, unlike cats. So when mathematicians realized they could square a number, they worked tirelessly to find a way to "unsquare" it, too. Alas, "unsquare" is not a real word. Instead, mathematicians (and now, your students) call this process "finding the square root."
The square root of 9 is written using something called the radical. We're not sure what's so radical about it, but it's probably better not to ask. What we do know is that it looks like this: . Likewise, "uncubing a number" is called "finding the cube root." Students should use the same radical for the cube root, only write a little 3 so we know it's a cube root and not a square root. For instance, the cube root of 27 can be written as .
Students should know that perfect squares and perfect cubes are integers that result from the squaring or cubing of another integer. For instance, 9 is a perfect square and 27 is a perfect cube because they can be written as 32 and 33, not because they're actually perfect. We don't want them to get a big head, now do we?
Students should understand that they can find the square root of any positive number and zero. (Imaginary numbers can wait until high school… unless you've got a couple of eager beavers whose heads you want to explode.) Unlike square roots, cube roots can be any number, positive or negative. It might be helpful to remind them that a negative number cubed is negative.
It's also important that students know the difference between rational and irrational numbers. They should be able to determine that the square root of a perfect square is rational and that other square roots like √2 and √3 are irrational.
While the answers to square roots can be both positive and negative (since either 22 or (-2)2 can equal 4), we'll only consider the principle (or positive) value of the square roots for our purposes.