The World of Numbers — Practice

Question 1

Is 0.272727… (recurring) rational or irrational?

Accepted Answer

It is non-terminating but recurring, so it is rational.

✦ Answer: rational.

Question 2

Simplify 2³ · 2².

Accepted Answer

Step 1 — same base, add exponents: 2³⁺² = 2⁵.

Step 2 — 2⁵ = 32.

✦ Answer: 32.

Question 3

Simplify √2 · √8.

Accepted Answer

Step 1 — √2 · √8 = √16.

Step 2 — √16 = 4.

✦ Answer: 4.

Question 4

Rationalise the denominator of 1/√5.

Accepted Answer

Step 1 — multiply by √5/√5.

Step 2 — (1·√5)/(√5·√5) = √5/5.

✦ Answer: √5/5.

Question 5

Find a rational number between 1/3 and 1/2.

Accepted Answer

Step 1 — take the average: (1/3 + 1/2)/2.

Step 2 — = (2/6 + 3/6)/2 = (5/6)/2 = 5/12.

✦ Answer: 5/12 (one of infinitely many).

Question 6

Express 0.4747… (47 recurring) as a fraction.

Accepted Answer

Step 1 — let x = 0.4747…; then 100x = 47.4747…

Step 2 — 100x − x = 47 → 99x = 47.

Step 3 — x = 47/99.

✦ Answer: 47/99.

Question 7

Rationalise the denominator of 1/(√7 − √3).

Accepted Answer

Step 1 — multiply by the conjugate (√7 + √3)/(√7 + √3).

Step 2 — denominator = (√7)² − (√3)² = 7 − 3 = 4.

Step 3 — result = (√7 + √3)/4.

✦ Answer: (√7 + √3)/4.

Question 8

Evaluate (64)^(2/3).

Accepted Answer

Step 1 — 64 = 4³, so (64)^(2/3) = (4³)^(2/3).

Step 2 — (4³)^(2/3) = 4^(3·2/3) = 4².

Step 3 — 4² = 16.

✦ Answer: 16.

Practice — The World of Numbers