# Lesson 6 - Simplifying Surds Rational Numbers and Integers

1.

Show that 0.45 is a rational number.

2.

Show that √9 ∈ ℤ .

3.

State whether each of the numbers are integers, rational numbers or irrational. Use ℤ, ℚ and ℝ to state the smallest number set each number is in. a.

7 b. −4.5

c. √2 d.

√5

2 e .

4.

Show that −√0.25 ∈ ℚ .

π f. √64

5.

Write the numbers below as fractions. (Note: The dots above digits implies recurrence. For example, 1. 1̇2̇ implies 1.121212…). a.

0. 1̇

,b. 0. 5̇

6.

Write 0. 0̇9̇ as a fraction.

Simplifying Surds – Easy

1.

√8 is equal to 2√2 . Why don’t we have to write ±2√2 ?

2.

Simplify the following: a.

√32 b. √20 c. √300 d. √72 e. √150 f. √96

3.

State whether each of the following are surds or not. a.

√11 b. √49 c. √50

4.

Simplify the following: a.

3√12 b. 2√9 c. 8√8 d.

14√18 e. 6√24 f. 7√54

5.

If we write a surd in its simplest form 𝑎√𝑏 , must 𝑏 be prime?

6.

When we find prime factors of a number, we divide out its prime numbers. When we simplify a surd, what kind of numbers do we divide out of the number being rooted?

Simplifying Surds – Hard

1.

Prove that √16 is not a surd. (Saying √𝟏𝟔 = ±𝟒 is not enough! Think about number sets and rules! Explain your process!)

2.

Simplify the following: a.

√192 b. √372 c. √484 d. √810

,e. √1728

,f. √1445

3.

If an expression 𝑎√𝑏 is not a surd where 𝑎 and 𝑏 are integers, write down the number set that 𝑎√𝑏 must be in. As usual, please write down the smallest number set it is contained in.

4.

Simplify the following: a.

5√60 b. 2√98

,c. 3√507

5.

For each of the roots, draw a prime factor tree and write the number inside the root as a product as its prime factors. Hence, simplify it. a.

√92 b. √288 c. √1875