GCSE Maths Indices, surds & standard form questions by topic

Use the index laws for this question.

1. (a)Write 2³ × 2² as a single power.[1]
2. (b)Write 7⁸ ÷ 7⁶ as a single power.[1]
3. (c)Work out (10²)³.[1]
4. (d)Work out 5⁰ + 5¹.[1]

Standard form conversions.

1. (a)Write 62 000 in standard form.[1]
2. (b)Write 3.4 × 10³ as an ordinary number.[1]
3. (c)Write 0.000 89 in standard form.[2]

Exact answers with π.

1. (a)Write down the exact circumference of a circle with diameter 14 cm.[2]
2. (b)Write down the exact area of a circle with radius 5 cm.[2]

Standard-form multiplication.

1. (a)Work out (2 × 10³) × (4 × 10⁴), giving your answer in standard form.[2]
2. (b)Work out (9 × 10⁶) ÷ (3 × 10²), giving your answer in standard form.[2]

A student is practising powers of ten.

1. (a)Write out 7 × 10⁴ in full, without using indices.[1]
2. (b)Convert 0.000 309 into standard form.[2]
3. (c)Simplify 10³ × 10⁴, leaving your answer as a power of ten.[1]

A cube and a square are compared.

1. (a)A cube has volume 216 cm³. Work out the length of one side.[2]
2. (b)A square has the same side length as the cube. Work out its area.[1]
3. (c)A second cube has sides 5 cm. Work out its volume.[1]

Mixed index simplification.

1. (a)Simplify (2x²y)³ × 4x⁴y⁵.[3]
2. (b)Simplify 12a⁸b³ ÷ 4a³b³.[2]

A hard drive stores data.

1. (a)A drive holds 5 × 10¹¹ bytes. A photo uses 2.5 × 10⁶ bytes. How many photos fit on the drive?[3]
2. (b)A second drive holds 1.6 × 10¹² bytes. How many times larger is it than the first drive?[2]

Exact values in a mixed calculation.

1. (a)Work out the exact value of √2 × √8.[2]
2. (b)A square has area 50 cm². Write the length of one side in the form a√b.[2]

Two quantities in standard form are compared.

1. (a)The mass of a whale is 1.5 × 10⁵ kg and the mass of a car is 1.2 × 10³ kg. How many cars balance one whale?[3]
2. (b)Write down one assumption behind your calculation.[1]

Light travels from the Sun to the Earth.

1. (a)Light travels 3 × 10⁵ km each second, and the Sun is 1.5 × 10⁸ km away. Work out the journey time in seconds, giving your answer in standard form.[2]
2. (b)Give the journey time in minutes, to 1 decimal place.[2]

A rectangle has sides √12 cm and √3 cm.

1. (a)Work out the exact area of the rectangle.[2]
2. (b)Show that the exact perimeter is 6√3 cm.[3]

Fractional and negative indices. (Higher)

1. (a)Work out 16^(3/4).[2]
2. (b)Work out 25^(−1/2).[2]
3. (c)Simplify (64x⁶)^(1/3). Give your answer as a single term.[2]

Rationalising denominators. (Higher)

1. (a)Rationalise 8⁄√2, giving your answer in simplest form.[2]
2. (b)Show that 5⁄(2 + √3) = 10 − 5√3.[4]

A surd shape problem. (Higher)

1. (a)A rectangle has sides (√8 + √2) cm and (√8 − √2) cm. Show that its area is 6 cm².[3]
2. (b)Find the perimeter of the rectangle, giving your answer as a simplified surd.[3]

Standard form in context. (Higher)

1. (a)Light travels at 3 × 10⁸ metres per second. Work out how far light travels in one day (86 400 seconds), giving your answer in standard form.[3]
2. (b)A star is 4.05 × 10¹⁶ metres away. How long does light take to reach us, in seconds? Give your answer as an ordinary number to 3 significant figures.[4]

This question is about index notation. (Higher)

1. (a)Solve 2^(2x) = 64.[2]
2. (b)Work out 27^(2/3) + 4^(3/2).[2]
3. (c)Simplify (3x²y)³ × 2x⁻⁵y⁻², giving your answer with positive indices.[3]

A data centre moves large amounts of data.

1. (a)The centre transfers 3.6 × 10¹² bytes every hour. Work out how many bytes it transfers each second, giving your answer in standard form.[3]
2. (b)A second site transfers 5 × 10¹¹ bytes every hour. Work out the two sites’ combined hourly total, in standard form.[3]