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Common Core Standards: Math

Expressions and Equations 8.EE.A.2

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.

Drills

1. Which of the following is rational?

Since 9 is a perfect square, its square root is the rational number 3. All other choices have irrational answers.

2. What is the value of ?

4

Since 4 squared is 16, it only makes sense that 4 is the square root of 16. The other options are incorrect because 22 = 4, 642 = 4,096, and 82 = 64. None of those are 16.

3. What is the value of ?

4

The cube root of 64 is the number that, when cubed, equals 64. The only number that equals 64 when cubed is 4 (because 4 × 4 × 4 = 64). Therefore, 4 is the cube root of 64. The other options are wrong because (A) equals 64 ÷ 3, (C) is the sixth root of 64 (because 26 = 64), and (D) is the square root, not cube root, of 64.

4. Which of the following has the greatest value?

We can calculate these roots relatively simply because they're all either perfect squares or perfect cubes. The value of (A) is -2 because (-2)3 = 8, the value of (B) is 1 because the square root of 1 is always 1, the value of (C) is 5 because 53 = 125, and the value of (D) is 8 because 82 = 64. So comparing -2, 1, 5, and 8, it's obvious that 8 has the greatest value.

5. What is the value of ?

10

Since 10 times itself is 100, so the square root of 100 is 10. Award yourself points if you got that question right.

6. Which of the following is rational?

Remember, the square root of perfect squares is going to be an integer, and integers are rational numbers. The perfect square here is 81, which is 92. So (D) is our answer. All other answers will eventually leave us with a factor of , which is irrational

7. Which of the following, when cubed, yields the largest answer?

1

When a negative number is cubed, the answer will be negative. Only choice (A) gives a positive answer because all the other options will be negative (-1, -8, and -125 when compared to 1).

8. Which of the following has both a rational square root and a rational cube root?

64

To find the answer, we can try finding the square and cube root of each answer choice and check if they're rational. The cube roots of (A) and (D) are both irrational, and the square root of (B) is irrational. Only (C) has a rational square root (82 = 64) and cube root (43 = 64).

9. What is ?

3

That question asks for the square root of the cube root of 729. (We know you were wondering how to say that.) First, we take the cube root of 729, which is 9. Then, we take the square root of that, which is 3. So our answer is (A).

10. What is the value of ?

25