IGCSE Maths Indices, surds, standard form topical questions

Answer without a calculator.

1. (a)Write down the value of 6² and of 4³.[1]
2. (b)Write down the value of √81.[1]
3. (c)Work out 2² × 2³.[2]

Convert between ordinary numbers and standard form.

1. (a)Write 605 000 in standard form.[1]
2. (b)Write 0.00000703 in standard form.[1]
3. (c)Write 5.1 × 10⁻³ as an ordinary number.[2]

Simplify each expression, leaving your answers as powers of 2.

1. (a)2⁶ ÷ 2²[1]
2. (b)(2²)³[1]
3. (c)2⁻¹ × 2⁵[1]

Simplify each expression.

1. (a)a⁷ × a³[1]
2. (b)b⁹ ÷ b⁴[1]
3. (c)(c⁵)³[1]
4. (d)6p² × 3p⁵[2]

This question is about standard form.

1. (a)Write 4.7 × 10⁴ as an ordinary number.[1]
2. (b)Write 9 × 10³ as an ordinary number.[1]
3. (c)Which is larger: 3.2 × 10⁵ or 29 500? Give a reason.[1]
4. (d)Write 91 000 000 in standard form.[2]

Work out in standard form.

1. (a)(3.2 × 10⁵) × (4 × 10³)[3]
2. (b)(8 × 10⁻⁵) ÷ (2 × 10⁻⁹)[2]

Evaluate exactly, without a calculator.

1. (a)5⁻²[1]
2. (b)2⁻³ × 2⁴ ÷ (2³)²[3]

A data centre stores 6.4 × 10⁸ files, and each file holds 2.5 × 10⁶ bytes. (Extended)

1. (a)Find the total number of bytes stored. Give your answer in standard form.[2]
2. (b)Another centre stores 4 × 10⁹ bytes. How many times larger is the first centre’s storage? Give your answer as an ordinary number.[2]

Answer without a calculator. (Extended)

1. (a)Evaluate 3⁻³.[1]
2. (b)Solve 5^x = 125.[1]
3. (c)Solve 2^x = 1/32.[2]
4. (d)Evaluate (3/4)⁻².[1]

Light travels at 3 × 10⁵ km per second. The Sun is about 1.5 × 10⁸ km from Earth.

1. (a)Find the time, in seconds, that sunlight takes to reach Earth.[2]
2. (b)Write this time in minutes and seconds.[1]
3. (c)One year is about 3.15 × 10⁷ seconds. Find the distance light travels in one year, in kilometres, giving your answer in standard form.[2]

Simplify, writing each answer in the form k√n where n is as small as possible. (Extended)

1. (a)√48[1]
2. (b)√50 + √18[2]
3. (c)√200 − √32[2]
4. (d)6/√3[1]

Rationalise the denominators. (Extended)

1. (a)10/√5[2]
2. (b)18/√6[2]
3. (c)1/(3 − √2)[3]

Answer this question without a calculator. (Extended)

1. (a)Expand and simplify (√7 + 2)(√7 − 3).[3]
2. (b)Show that (3 + √5)(3 − √5) = 4.[2]
3. (c)Write 15/√3 with a rational denominator, simplifying the surd fully.[2]

Answer this question without a calculator. (Extended)

1. (a)Given 8^x = 2¹², find the value of x.[2]
2. (b)Evaluate (2⁻¹ + 3⁻¹)⁻¹.[3]
3. (c)Show that 27^(2/3) ÷ 9^(1/2) = 3.[2]