Simplifying Surds

1. Simplifying Surds Slideshow 6, Mr Richard Sasaki, Room 307

2. Objectives • Understand the meaning of rational numbers • Understand the meaning of surd • Be able to check whether a number is a surd or not • Be able to simplify surds

3. Rationality First we need to understand the meaning of rational numbers. What is a rational number? A rational number is a number that can be written in the form of a fraction. is rational if where . If where , we say . ( is in the rational number set, .)

4. Rationality irrational If a number is not rational, we say that it is . is irrational if it can’t be written in the form where . Therefore, an irrational number . Example Show that 0.8 . . Note: If , .

5. Answers and . . where . . where -3 and 3 are integers. . Let and Let and Let and

6. Surds What is a surd? A surd is an irrational root of an integer. We can’t remove its root symbol by simplifying it. Are the following surds? Yes! Yes! Yes! No! Even if the expression is not fully simplified, if it is a root and irrational, it is a surd.

7. Multiplying Roots How do we multiply square roots? Let’s consider two roots, and where . If we square both sides, we get… If we square root both sides, we get… , where .

8. Simplifying Surds To simplify a surd, we need to write it in the form where is as small as possible and . Note: Obviously, if . Example Simplify . We try to take remove square factors out and simplify them by removing their square root symbol.

9. Answers - Easy Because has positive and negative roots anyway. . No, of course not! is a surd but 6 is not prime. Square numbers.

10. Answers - Hard Let be in the form where . simplifies to or rather in the form . We can see , hence is not a surd. 1875 288 92 625 ③ 144 46 ② ② 23 72 ② ② 125 ⑤ 25 ⑤ 36 ② 18 ⑤ ② ⑤ 9 ② ③ ③