1ST LESSON ON RADICALS - Understanding Squares, Radicands, and Rational Numbers

HERE ARE THE FIRST TEN square numbers and their roots:

Square numbers 1 4 9 16 25 36 49 64 81 100

Square roots of those same numbers 1 2 3 4 5 6 7 8 9 10

We write, for example,

R25 = 5.

"The square root of 25 is 5."

This mark is called the radical sign (after the Latin radix = root). The number under the radical sign is called the radicand. In the example, 25 is the radicand.

Problem 1. Evaluate the following.

a) R64= ____
b) R144= _____
c) R400=_____
d) R289= _____
e) R______= 1
f) R______= 7


Example 1. Evaluate Radicand over 13 x 13.

Solution = 13

For, 13 x 13 is a square number. And the square root of 13 x 13 is 13.

If a is any positive number, then a x a is obviously a square number, and

Radicand over a x a = a

Problem 2. Evaluate the following.

a) Radicand over 28 x 28= ______
b) Radicand over 135 x 135= ________
c) Radicand over __________= 2· 3· 5 = 30.

We can state the following theorem:

A square number times a square number is itself a square number.

For example,

36 x 81 = 6 x 6 x 9 x 9 = 6 x 9 x 6 x 9 = 54 x 54

Problem 3. Without multiplying the given square numbers, each product of square numbers is equal to what square number? I did the first one for you.

a) 25 x 64 = Solution 5 x 8 x 5 x 8 = 40 x 40

b) 16 x 49 =

c) 4 x 9 x 25 =

Rational and irrational numbers

The rational numbers are the numbers of arithmetic: the whole numbers, fractions, mixed numbers, and decimals; together with their negative images.

That is what a rational number is. As for what it looks like, it will take the form a / b, where a and b are integers and
(b ≠ 0).

Problem 4. Which of these are rational or irrational numbers?

1. _____1
2. _____−6
3. _____3½
4. _____4
5. _____−13
6. _____0
7. _____-13 / 5
8. _____7.38609

At this point, the student might wonder, What is a number that is not rational?

An example of such a number is R2 or("Square root of 2"). R2 is not a number of arithmetic. There is no whole number, no fraction, and no decimal whose square is 2. 7 / 5 is close, because 7 / 5 x 7 / 5 = 49 / 25
-- which is almost 2.

But to prove that there is no rational number whose square is 2, then suppose there were. Then we could express it as a fraction m / n in the lowest terms. That is, suppose m / n x m / n = m x m / n x n = 2. But that is impossible. Because since m / n is in lowest terms, then m and n have no common divisors except 1.

Therefore, m x m and n x n also have no common divisors -- they are relatively prime -- and it will be impossible to divide n x n into m x m and get 2!

There is no rational number whose square is 2. Therefore we call the R2(square root of 2) an irrational number.

Question. Which square roots are rational?
Answer: Only the square roots of square numbers.

R1 = 1 Rational

R2 Irrational

R3 Irrational

R4 = 2 Rational

R5, R6, R7, Irrational

R9 = 3 Rational

And so on.

Only the square roots of square numbers are rational.

The existence of these irrationals was first realized by Pythagoras in the 6th century B.C. He called them "unnameable""speechless" numbers. For, if we ask, "How much is R2? -- we cannot say. We can only call it, R2 or "Square root of 2."

Problem 5. Say the name of each number.

a) R2___________________
b) R8___________________
c) R9___________________
d) R4 / 25 _________________
e) R10 __________________

Good Job Cassidy....Hopefully this helps. I will post another lesson after this!!