GCSE Maths Powers, Roots & Fractional Indices Worksheet PDF

GCSE Maths – Number Powers, Roots and Fractional Indices Worksheet NOTES SOLUTIONS This worksheet will show you how to work out different types of questions on powers, roots and fractional indices. Eac...

GCSE Maths – Number Powers, Roots and Fractional Indices Worksheet NOTES SOLUTIONS This worksheet will show you how to work out different types of questions on powers, roots and fractional indices. Each section contains a worked example, a question with hints and then questions for you to work through on your own. This work by PMT Education is licensed under https://bit.ly/pmt-cc https://bit.ly/pmt-edu-cc CC BY-NC-ND 4.0 https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Section A Worked Example 1 Find 13 3 Step 1: Identify the power and use the number to indicate how many multiples there are. 𝟏𝟏𝟑𝟑𝟑𝟑 = 𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 Step 2: Calculate the product. 𝟏𝟏𝟑𝟑𝟑𝟑 = 𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 × 𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐 Worked Example 2 𝟓𝟓 −𝟒𝟒 Find 𝟒𝟒 Step 1: Due to the negative sign, flip the base. 5 −4 4 4 = 4 5 Step 2: Apply the remaining power to the numerator and denominator. 5 −4 4 4 44 4 × 4 × 4 × 4 256 = = 4= = 4 5 5 5 × 5 × 5 × 5 625 Guided Example 1 Find 21 2 Step 1: Identify the power and use the number to indicate how many multiples there are. Step 2: Calculate the product. Guided Example 2 𝟏𝟏𝟏𝟏 −𝟐𝟐 Find 𝟑𝟑 Step 1: Due to the negative sign, flip the base. Step 2: Apply the remaining power to the numerator and denominator. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Now it’s your turn! If you get stuck, look back at the worked and guided examples. 1. Find 73 2. Find 44 3. Find 56 3 2 4. Find 5 −9 3 5. Find 4 6. Find 2−3 7. Find 0.55 7 −4 8. Find 11 2 7 9. Find 3 10. Find (−11)−4 https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Section B Worked Example 1 5 Simplify 𝑔𝑔 × 𝑔𝑔 3 Step 1: As we are multiplying, we must add the two powers together. 𝑔𝑔5 × 𝑔𝑔3 = 𝑔𝑔5+3 Step 2: Simplify the addition of the two powers. 𝑔𝑔5+3 = 𝑔𝑔8 Worked Example 2 6 11 Simplify (𝑞𝑞 ) Step 1: As we are raising a power to another power, we must multiply the two powers together. (𝑞𝑞 6 )11 = 𝑞𝑞 6×11 Step 2: Simplify the multiplication of the two powers. 𝑞𝑞 6×11 = 𝑞𝑞 66 Guided Example 𝟏𝟏 Simplify 𝒚𝒚𝟗𝟗 ÷ 𝒚𝒚 𝟐𝟐 Step 1: As we are dividing, we must subtract the second powers from the first power. Step 2: Simplify the subtraction of the two powers. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Now it’s your turn! If you get stuck, look back at the worked and guided examples. 11. Simplify 𝑥𝑥 × 𝑥𝑥 × 𝑥𝑥 12. Simplify 𝑎𝑎3 × 𝑎𝑎4 13. Simplify 𝑟𝑟 40 ÷ 𝑟𝑟 21 3 1 14. Simplify 𝑒𝑒 4 × 𝑒𝑒 2 7 15. Simplify 𝑡𝑡 3 ÷ 𝑡𝑡 2 16. Simplify (𝑎𝑎2 )3 17. Simplify (9𝑏𝑏 4 )7 9 18. Simplify (3𝑓𝑓 5 )10 19. Simplify (𝑝𝑝−𝑞𝑞 )−𝑟𝑟 𝑥𝑥 2𝑦𝑦 3 20. Simplify ( ) 𝑥𝑥 𝑦𝑦 https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Section C – Higher Only Worked Example 1 Find and simplify √𝟔𝟔𝟔𝟔 Step 1: Identify if the number in the root has any square number factors. 68 = 4 × 17 so 4 is a square number factor. Step 2: Simplify the square root using rules of surds. √68 = √4 × 17 = √4 × √17 = 2 × √17 = 𝟐𝟐√𝟏𝟏𝟏𝟏 Worked Example 2 𝟒𝟒 Find and simplify √𝟔𝟔𝟔𝟔𝟔𝟔 Step 1: Without using a calculator, find an integer which factors into 625 exactly 4 times (the same number of times as the root). 625 = 5 × 5 × 5 × 5 = 54 Step 2: Deduce the solution to the root expression. 4 4 √625 = 54 = 𝟓𝟓 Guided Example 1 Find and simplify √𝟏𝟏𝟏𝟏𝟏𝟏 Step 1: Identify if the number in the root has any square number factors. Step 2: Simplify the square root using rules of surds. Guided Example 2 𝟓𝟓 Find and simplify √𝟑𝟑𝟑𝟑 Step 1: Without using a calculator, find an integer which factors into 32 exactly 5 times (the same number of times as the root). Step 2: Deduce the solution to the root expression. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Now it’s your turn! If you get stuck, look back at the worked and guided examples. 21. Find √81 22. Find √24 23. Find √900 24. Find √612 25. Find 2√128 26. Find 13√338 3 27. Find √64 4 28. Find √16 3 29. Find √125 5 30. Find √243 https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Section D – Higher Only Worked Example 𝟑𝟑 Find and simplify 𝟐𝟐− 𝟐𝟐 Step 1: Due to the negative sign, flip the base. 3 3 1 2 2− 2 = 2 Step 2: Apply the remaining index to the numerator and denominator. 3 3 1 2 12 = 3 2 22 Step 3: Simplify the remaining powers and roots. Rationalise the denominator if necessary. 3 12 1 1 1 √8 √8 √4 × 2 √4 × √2 2√2 √2 3 = = = × = = = = = √23 √8 √8 √8 8 8 8 8 4 22 Guided Example 𝟏𝟏 𝟒𝟒 Find and simplify 𝟐𝟐 𝟓𝟓 Step 1: Apply the power to the numerator and denominator. As it is a fraction, we will get a root. Step 2: Simplify the remaining powers and roots. Rationalise the denominator if necessary. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Now it’s your turn! If you get stuck, look back at the worked and guided examples. 1 31. Find and simplify where possible 92 3 32. Find and simplify where possible 12−2 5 4 2 33. Find and simplify where possible 3 1 27 −3 34. Find and simplify where possible 64 −12 35. Find and simplify where possible 9 https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  2 7 −3 36. Find and simplify where possible 8 5 16 −4 37. Find and simplify where possible 81 4 8 −3 38. Find and simplify where possible 27 3 9 −2 39. Find and simplify where possible 16 2 40. Find and simplify where possible (32)−5 https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Section E – Higher Only Worked Example Estimate 6. 5 2 Step 1: Recognise that 6.5 is between two integers, 6 and 7. 6 < 6.5 < 7 Step 2: Due to this, 6. 52 is between 62 and 72. 62 < 6. 52 < 72 Step 3: Simplify this inequality. 36 < 6. 52 < 49 Step 3: Using this we can estimate 6. 52. 6. 52 ≃ 40 Guided Example Estimate √𝟏𝟏𝟏𝟏 Step 1: Recognise that 14 is between two square numbers, 9 and 16. Step 2: Due to this the square root of 14 lies between the square root of 9 and the square root of 16. Step 3: As 14 is closer to 16 than to 9, square root of 14 is close to the square root of 16. https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc  Now it’s your turn! If you get stuck, look back at the worked and guided examples. 41. Estimate 4. 32 42. Estimate 1. 43 43. Estimate 2. 15 44. Estimate 0. 8232 45. Estimate √39 46. Estimate √35 47. Estimate √140 48. Estimate √18.2 3 49. Estimate √61 https://bit.ly/pmt-cc https://bit.ly/pmt-edu https://bit.ly/pmt-cc