Surds and Indices Worksheet — GCSE — Advanced

Surds and Indices worksheet for Key Stage 4 / GCSE (Ages 14-16). Advanced level maths practice, aligned to the UK National Curriculum. Print-ready with answer key included.

Maths•Key Stage 4 / GCSE (Ages 14-16)•Surds and Indicesadvanced

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Mastering Surds and Indices

Key Stage 4 / GCSE (Ages 14-16) - Advanced Level

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Instructions: This worksheet is designed to enhance your understanding of surds and indices through a series of challenging problems. Carefully read each question and show all your workings where necessary.

Simplifying Surds

Simplify each of the following surds. Show all steps clearly to ensure full marks.

1. √50

2. √72

3. √98

4. √200

5. √128

Rationalising the Denominator

Rationalise the denominator of each expression and simplify your answer fully.

1. 5/√2

2. 3/(2 + √3)

3. 7/(√5 - 1)

4. 4/(3√2)

5. 2/(√7 + √2)

Laws of Indices

Apply the laws of indices to simplify the following expressions. Provide your answers in their simplest form.

1. (x^3 * x^4)

2. (y^5 / y^2)

3. (a^2)^3

4. (b^4 * b^-2)

5. (c^0)

Solving Equations with Indices

Solve the following equations involving indices. Show all steps in your working.

1. 2^(x+1) = 16

2. 3^(2x) = 81

3. 5^(x-2) = 1/25

4. 4^(3x) = 64

5. 7^(x+2) = 343

Comparing Surds and Indices

Compare the following pairs of expressions and determine which is larger. Provide justification for your answer.

1. √45 or 6.7

2. 2√3 or 3.5

3. 3^(1/2) or 1.7

4. 5^(1/3) or 1.7

5. √12 or 3.4

Indices in Context

Use your knowledge of indices to solve these real-world problems. Show all calculations clearly.

1. If a population of bacteria doubles every hour, express the population after 5 hours if the initial population is 100.

2. A car depreciates by 10% each year. If its initial value is £20,000, what will its value be after 3 years?

Surds in Geometry

Solve these geometry problems involving surds. Round your final answers to two decimal places if necessary.

1. The diagonal of a square is √50 cm. Find the length of one side of the square.

2. A right-angled triangle has sides of length 7 cm and 24 cm. Find the length of the hypotenuse in surd form.

Extended Writing: Surds and Indices

Write a detailed essay explaining the importance of surds and indices in mathematics. Discuss their applications in various fields and provide examples to support your discussion.

Discuss the significance of surds and indices in real-world applications, including engineering and science. Provide examples of how these concepts are used in calculations and problem-solving. [15 marks]

Answer Key

Activity 1: 5√2 ; 6√2 ; 7√2 ; 10√2 ; 8√2

Activity 2: 5√2/2 ; (3(2-√3))/1 ; 7(√5+1)/4 ; 4√2/6 ; 2(√7-√2)/5

Activity 3: x^7 ; y^3 ; a^6 ; b^2 ; 1

Activity 4: x = 3 ; x = 2 ; x = 4 ; x = 2 ; x = 1

Activity 5: 6.7 ; 3.5 ; 3^(1/2) ; 5^(1/3) ; 3.4

Activity 6: 3200 ; £14,580

Activity 7: 5√2 cm ; 25 cm

Activity 8: Essay will vary