Indices and Standard Form

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1 Worksheets for GCSE Mathematics Indices and Standard Form Mr Black Maths Resources for Teachers GCSE 1-9 Number

2 Indices and Standard Index Form Worksheets Contents Differentiated Independent Learning Worksheets Square and Cube Numbers Page 20 Index Notation Page 30 Rules of Indices Page 40 Writing with Standard Index Form Page 50 Calculations with Standard Index Form Page 60 Simplifying Surds Page 70 Calculations with Surds Page 80 Negative Powers & Reciprocals Page 90 Fractional Powers Page 10 Rationalising the Denominator Page 11 Solutions Square and Cube Numbers Page 12 Index Notation Page 13 Rules of Indices Page 14 Writing with Standard Index Form Page 15 Calculations with Standard Index Form Page 16 Simplifying Surds Page 17 Calculations with Surds Page 18 Negative Powers & Reciprocals Page 19 Fractional Powers Page 20 Rationalising the Denominator Page 21 1

3 Square and Cube Numbers Q1. Calculate the following square numbers and square roots. a) 3 b) 5 c) 1 d) 4 e) 12 g) 100 h) 36 i) 64 j) 49 k) 169 m) n) o) 3 2 p) q) 5 3 Q2. Use a calculator to evaluate the following. Round your answers to two decimal places when necessary. a) 15 b) 20 c) d) e) f) 900 g) 2 h) 39 i) 91 j) 4493 Q3. Calculate the following square numbers and square roots. a) 3 b) 1 c) 5 d) 27 e) 64 Q4. Use a calculator to evaluate the following. a) 8 b) 14 c) 7 2 d) e) 729 f) 1331 g) 2197 h) Q5. Arrange these calculations in ascending order. a) b) Q6. Use square and cube numbers to complete these calculations. a) + =106 b) + =164 c) =40 d) + =91 e) + =72 f) + =31 2

4 Index Notation Q1. Write the following expressions using a single power. a) 5 5 b) 4 4 c) d) 2 2 e) f) g) h) 2 2 i) 5 5 j) 7 7 k) l) Q2. Write the following expressions using a single power. a) b) c) d) e) f) g) j) h) i) k) l) Q3. Write the following expressions using a single power. a) b) c) d) e) Q4. Match together the following expressions with their value of. 3 3 =3 =2 = = =5 =3 = = 2 = = 1 2 3

5 Rules of Indices Q1. Use the rules of indices to simplify the following expressions. a) 3 5 b) 4 5 c) 6 4 d) 15 3 e) 6 2 f) g) 8 4 h) 6 4 i) 18 9 j) k) l) Q2. Use the rules of indices to simplify the following expressions. a) 3 6 b) 5 2 c) 5 4 d) 12 4 e) f) g) h) i) Q3. Complete the indices multiplication grids r Q4. Write an expression for the area of these shapes. a) b) 4

6 Standard Index Form Q1. Convert these ordinary numbers to standard index form. a) 6000 b) c) 700 d) 90 e) 3450 f) 8500 g) h) 26 i) j) k) l) m) 5 million n) 6.2 billion o) p) Q2. Convert these ordinary numbers to standard index form. a) 0.03 b) 0.10 c) d) e) f) g) h) i) 0.8 j) k) l) m) n) o) p) Q3. Convert these numbers from standard index form to ordinary form. a) b) 6 10 c) d) e) f) g) h) i) 7 10 j) k) l) m) n) o) p) Q4. Which of these numbers is the odd one out? Explain your decision. a) Q5. Here are the areas of some of world s largest countries. Give your answers in standard form. i) What is the total area for Russia and Canada? ii) What is the difference in area between China and the United States? iii) What is the total area of the five countries? Russia km 2 Canada km 2 China km 2 United States km 2 5

7 Standard Index Form Problems Q1. Calculate the following. Leave your answers in standard index form. Do not use a calculator. a) b) c) d) e) f) Q2. Calculate the following. Leave your answers in standard index form. Do not use a calculator. a) b) c) d) e) f) Q3. Calculate the following. Leave your answers in standard index form. Do not use a calculator. a) d) b) e). c) f).. Q4. Use a calculator to calculate the following. Give your answer correct to 3 significant figures. a) b) c) d) e)... f)... Q5. The radius of the Earth is approximately m. Calculate the volume of the Earth given that =. Q6. The mass of the moon is Kg. The Earth has a mass 81 times that of the moon. a) Calculate the mass of the Earth. The mass of Mars is Kg. b) Calculate the difference between the mass of the moon and the mass of Mars. Q7. Singapore has a population of and an area of 710 Km 2. Bangladesh has a population of with an area of a) Calculate the population density of each country. b) What is the difference between the population density of Singapore and Bangladesh? 6

8 Simplifying Surds Q1. Simplify the following surds leaving your answers in the form. a) 12 b) 40 c) 200 d) 72 e) 27 f) 75 g) 90 h) 150 i) 128 j) 98 k) 63 l) 44 Q2. Simplifying the following leaving your answers in the form. a) 4 10 b) 6 8 c) 3 18 d) e) 24 2 f) g) h) 3+ 3 i) j) k) l) m) n) o) Q3. Simplify the following surds as fully as possible. a) b) c) d) e) Q4. Match together the equivalent values f)

9 Calculations with Surds Q1. Multiply out the brackets and simplify. a) b) c) d) e) f) Q2. Multiply out the brackets and simplify. a) b) c) d) e) f) g) h) i) Q3. Multiply out the brackets and simplify. a) 3+5 b) c) d) 6 3 e) f) Q4. a) If =4+ 3 and =4 2 3 simplify: i) + ii) iii) b) If =5 6 and =1+2 2 simplify: i) 5 ii) iii) 2 Q5. a) Write an exact expression for the area of the shaded region. Fully simplify your answer b) The shaded square has been constructed by joining the mid-points of a larger square. Write the perimeter of the larger square in surd form. 10 8

10 Reciprocals Q1. Calculate the reciprocal of the following. Leave your answers as simplified fractions. a) 3 b) 8 c) 10 d) 0.4 e) 0.25 f) 0.01 g) h) i) 2.1 j) k) 1 l) 3 Q2. Evaluate the following. Leave your answers as simplified fractions. a) 7 b) 0.2 c) 1.5 d) 2.6 e) f) 1 Q3. Calculate the reciprocal of each term to complete the table Q4. Evaluate the following powers. Leave your answer as a simplified fraction. a) 3 b) 5 c) 6 d) 2 e) 2 f) 1 g) 8 h) 3 i) 5 j) k) l) m) n) 1 o) 1 Q5. Arrange the following cards in ascending order

11 Q1. Write the following using index notation. Fractional and Negative Powers a) 34 b) 4 e) 5 f) 6 c) 10 g) 3 d) 200 h) 4 i) 5 j) 143 k) 726 l) 145 Q2. Evaluate the following. Give your answers as whole numbers or fractions. a) 4 d) 27 g) 36 j) 125 b) 9 e) 49 h) 64 k) 81 c) 16 f) 4 i) 27 l) 125 Q3. Arrange the following in ascending order Q4. Match the fractional power to the base. 27= 81 4=16 64=32 25= Q5. Evaluate the following. Give your answers as whole numbers or fractions. a) b) c) d) 10

12 Q1. Rationalise the following denominators. Rationalising the Denominator a) b) c) d) e) f) g) h) i) j) k) l) Q2. Rationalise the following denominators. a) b) c) d) e) f) g) h) Q3. Rationalise the denominator and simplify each fraction. a) d) b) e) c) f) Q4. Match together the following cards Q5. a) b) c) Q6. Calculate

13 Solutions Square and Cube Numbers Q1. a) 9 b) 25 c) 1 d) 16 e) 144 g) 10 h) 6 i) 8 j) 7 k) 13 m) 34 n) 68 o) 5 p) 5 q) 4 Q2. a) 225 b) 400 c) d) e) f) 30 g) 1.41 h) 6.25 i) 9.54 j) Q3. a) 27 b) 1 c) 125 d) 3 e) 4 Q4. a) 512 b) 2744 c) 335 d) 1216 e) 9 f) 11 g) 13 h) 22 Q5. a) b) Q6. Use square and cube numbers to complete these calculations. a) 9 +5 =106 b) =164 c) 7 3 =40 d) 3 +4 =91 e) 8 +2 =72 f) 3 +2 =31 12

14 Index Notation Q1. a) 5 b) 4 c) 7 d) 2 e) 6 f) 9 g) 3 h) 2 i) 5 j) 7 k) 8 l) 3 Q2. a) b) c) d). e) f) g) h) i) j) 1 k) l) Q3. a) b) c) d) e) Q =3 =2 = = =5 =3 = = 2 = =

15 Rules of Indices Q1. a) 15 b) 20 c) 24 d) 5 e) 3 f) 2 g) 2 h) 24 i) 2 j) 8 k) 8 l) 4 Q2. a) 18 b) 10 c) 20 d) 3 h e) 28 f) g) h) 2 i) Q r Q4. a) = 12h b) = 15 14

16 Standard Index Form Q1. a) 6 10 b) 4 10 c) 7 10 d) 9 10 e) f) g) h) i) j) k) l) m) 5 10 n) o) p) Q2. a) 3 10 b) 1 10 c) 4 10 d) e) f) g) h) i) 8 10 j) 5 10 k) 1 10 l) m) n) o) p) Q3. a) b) 600 c) 9.44 d) e) 2560 f) 6392 g) h) i) 0.07 j) 0.12 k) l) m) n) o) p) Q is the odd one out since it is not written using correct index form. Q5. i) km 2 ii) 6 10 km 2 iii) km 2 15

17 Standard Index Form Problems Q1. a) 8 10 b) 6 10 c) 6 10 d) 2 e) 2100 f) Q2. a) 3000 b) 30 c) 2 10 d) 2 10 e) 700 f) 200 Q3. a) b) 8 10 c) 12 d) e) f) 6 Q4. a) b) c) d) e) f) Q5. = m Q6. a) Mass of the Earth = b) Kg Q7. a) Singapore population density = 7301 people/km 2 Bangladesh population density = 1027 people/km 2 b) Difference = 6274 people/km 2 16

18 Simplifying Surds Q1. a) 2 3 b) 2 10 c) 10 2 d) 6 2 e) 3 3 f) 5 3 g) 3 10 h) 5 6 i) 8 2 j) 7 2 k) 3 7 l) 2 11 Q2. a) 2 10 b) 4 3 c) 3 6 d) e) 2 3 f) 2 5 g) 2 5 h) 2 3 i) 5 2 j) k) 6 3 l) 11 3 m) 5 5 n) 2 2 o) 9 5 Q3. a) 1 b) 8 3 c) 2 14 d) 0 e) f) Q

19 Calculations with Surds Q1. a) b) c) d) e) f) Q2. a) b) c) 2 18 d) e) f) 7 30 g) 13 h) 6 i) 4 Q3. a) b) c) d) e) f) Q4. a) i) 8 3 ii) 3 3 iii) b) i) ii) iii) Q5. a) Area = units 2 b) Perimeter = 8 5 units 18

20 Reciprocals Q1. a) b) c) d) e) 4 f) 100 g) h) i) j) k) l) Q2. a) b) 5 c) d) e) 5 f) Q Q4. a) d) b) 5 e) c) 6 f) 1 g) h) i) j) 4 k) 27 l) m) n) o) Q

21 Fractional and Negative Powers Q1. a) 34 b) 4 c) 10 d) 200 e) 5 f) 6 g) 3 h) 4 i) 5 j) 143 k) 726 l) 145 Q2. a) 2 b) 3 c) 4 d) 9 e) f) 32 g) 36 h) 216 i) j) k) l) Q3. a) =3,=4 b) =2,=1 c) =5,=4 d) =3,=2 Q4. a) b) c) d) 20

22 Rationalising the Denominator Q1. a) b) c) d) e) f) g) h) i) j) k) l) Q2. a) b) c) d) e) f) 2 g) h) Q3. a) +1 b) +1 c) d) e) Q4. f) Q5. a) b) c) -1 Q6. 21

Worksheets for GCSE Mathematics. Accuracy & Rounding. Mr Black's Maths Resources for Teachers. Number

Worksheets for GCSE Mathematics Accuracy & Rounding Mr Black's Maths Resources for Teachers Number Accuracy & Rounding Worksheets Contents Differentiated Independent Learning Worksheets Rounding to the