Number

Simplify √50.

• A10√5
• B25√2
• C2√5
• D5√2✓

Answer: √50 = √(25×2) = √25 × √2 = 5√2.

Write 0.00056 in standard form.

• A5.6 × 10⁴
• B5.6 × 10⁻³
• C5.6 × 10⁻⁴✓
• D56 × 10⁻⁵

Answer: Move the decimal point 4 places right to get 5.6, so 0.00056 = 5.6 × 10⁻⁴.

Evaluate 5⁰.

• Aundefined
• B1✓
• C5
• D0

Answer: Any non-zero number raised to the power 0 equals 1.

Evaluate 16^(1/2).

• A32
• B8
• C256
• D4✓

Answer: A power of ½ means the square root: 16^(1/2) = √16 = 4.

Simplify 2⁵ × 2³.

• A2²
• B2¹⁵
• C2⁸✓
• D4⁸

Answer: When multiplying powers of the same base, add the indices: 2⁵ × 2³ = 2⁸.

Rationalise the denominator of 1/√2.

• A√2
• B√2/2✓
• C√2/4
• D1/2

Answer: Multiply top and bottom by √2: 1/√2 × √2/√2 = √2/2.

Express 0.4 recurring (0.444…) as a fraction.

• A4/10
• B4/99
• C40/99
• D4/9✓

Answer: Let x = 0.444…; 10x = 4.444…; subtracting gives 9x = 4, so x = 4/9.

What is 3/4 + 2/3?

• A5/12
• B6/12
• C17/12✓
• D5/7

Answer: Common denominator 12: 9/12 + 8/12 = 17/12 (= 1 5/12).

Use algebra to write the recurring decimal 0.454545… as a fraction in its simplest form.

• A41/90
• B5/11✓
• C9/20
• D5/99

Answer: Let x = 0.454545…. Two digits recur, so 100x = 45.454545…. Subtracting gives 99x = 45, so x = 45/99, and dividing top and bottom by 9 gives 5/11.

Expand and simplify (3 + √5)(2 − √5), giving your answer in the form a + b√5.

• A1 + √5
• B1 − 5√5
• C1 − √5✓
• D11 − √5

Answer: Expanding gives 6 − 3√5 + 2√5 − (√5)². Since (√5)² = 5, this is 6 − 5 − √5 = 1 − √5. The two surd terms combine to −√5, not −5√5.

Rationalise the denominator of 6/(3 − √2) and simplify.

• A(18 − 6√2)/7
• B(18 + 6√2)/11
• C3 + √2
• D(18 + 6√2)/7✓

Answer: Multiply top and bottom by the conjugate 3 + √2. The denominator becomes 3² − (√2)² = 9 − 2 = 7 and the numerator becomes 6(3 + √2) = 18 + 6√2, giving (18 + 6√2)/7.

A length L is measured as 8.4 cm, correct to 1 decimal place. Write down the error interval for L.

• A8.3 ≤ L < 8.5
• B8.35 < L ≤ 8.45
• C8.35 ≤ L < 8.45✓
• D8.39 ≤ L < 8.41

Answer: Values from half of 0.1 below to half of 0.1 above round to 8.4, so the bounds are 8.35 and 8.45. A value of exactly 8.35 rounds up to 8.4 and is included, while 8.45 rounds to 8.5 and is excluded.

A car travels 240 m, measured to the nearest 10 m, in 12.0 s, measured to the nearest 0.1 s. Work out the upper bound for the average speed, in m/s, to 3 significant figures.

• A19.7 m/s
• B20.0 m/s
• C20.3 m/s
• D20.5 m/s✓

Answer: Speed is largest when the distance is largest and the time is smallest, so use 245 m and 11.95 s. 245 ÷ 11.95 = 20.5 m/s to 3 significant figures.

A plank is 2.60 m long, measured to the nearest centimetre. A piece 0.85 m long, also measured to the nearest centimetre, is cut from it. Work out the lower bound of the length of plank that remains.

• A1.740 m✓
• B1.745 m
• C1.750 m
• D1.760 m

Answer: What is left is smallest when the plank is as short as possible and the piece removed is as long as possible, so use 2.595 − 0.855 = 1.740 m. Taking the lower bound of both lengths would make the remainder too large.

Work out the exact value of 27^(−2/3).

• A9
• B1/9✓
• C1/729
• D−9

Answer: The denominator of the fractional power gives the cube root of 27, which is 3, and the numerator squares it to give 9. The negative power then takes the reciprocal, giving 1/9; a negative index does not make the answer negative.

Simplify √48 + √75 − √27, giving your answer in the form a√3.

• A6√3✓
• B2√3
• C4√6
• D12√3

Answer: √48 = 4√3, √75 = 5√3 and √27 = 3√3. Since all three are multiples of √3 they combine directly: 4 + 5 − 3 = 6, giving 6√3. Roots cannot be added by adding the numbers underneath them.