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## Powers & Roots

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**Powers & Roots**Squares, Cubes and Powers (Indices) Square Roots and Cube Roots Large Numbers Scientific Notation www.mathsrevision.com Small Numbers Scientific Notation Scientific Notation using Calculator Compiled by Mr. Lafferty Maths Dept.**Starter Questions**www.mathsrevision.com Compiled by Mr. Lafferty Maths Dept.**Squaring a Number**Learning Intention Success Criteria • To understand what is meant by the term power of a number. • We are learning the term • about powers of a number’. • Be able to calculate powers of a number squares both mentally and using the calculator. www.mathsrevision.com Compiled by Mr. Lafferty Maths Dept.**Square of a Number**To square a number means to : “Multiply it by itself” Example : 92 means 9 x 9 = 81 www.mathsrevision.com 102 means 10 x 10 = 100 Compiled by Mr. Lafferty Maths Dept.**Cube of a Number**To cube a number means to : “Multiply it by itself and then itself again” Example : 23 means 2 x 2 x 2 = 8 www.mathsrevision.com 43 means 4 x 4 x 4 = 64 Compiled by Mr. Lafferty Maths Dept.**Index of a Number**n is called the index number an a is called the base number 25 Example : www.mathsrevision.com 32 means 2 x 2 x 2 x 2 x 2 = Compiled by Mr. Lafferty Maths Dept.**NON - CALCULATOR**9 1 0 0 0 -8 2 1 6 2 5 6 1 Mr. Lafferty@mathsrevision.com**Area of Square**Area of a square is l cm A = l x l = l2 Example : Find the area of a square with length 7cm. www.mathsrevision.com A = l2 = 72 = 49cm2 Compiled by Mr. Lafferty Maths Dept.**Special button**on the calculator Compiled by Mr. Lafferty Maths Dept.**Powers & Roots**Now try TJ4+ Ex 10.1 Ch10 (page 64) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.**Square Root of a Number**92 = 9 x 9 = 81 You now know how to find : We can ‘undo’ this by asking “which number, times itself, gives 81” From the top line, the answer is 9 www.mathsrevision.com This is expressed as : “the SQUARE ROOT of 81 is 9” or in symbols we write : Compiled by Mr. Lafferty Maths Dept.**Cube Root of a Number**23 = 2 x 2 x 2 = 8 You now know how to find : We can ‘undo’ this by asking “which number, times itself, times itself again gives 8” www.mathsrevision.com From the top line, the answer is 2 This is expressed as : “the CUBE ROOT of 8 is 2” or in symbols we write : Compiled by Mr. Lafferty Maths Dept.**Roots of a Number**Example : 42 means 4 x 4 = 16 so √16 = 4 53 means 5 x 5 x 5 = 125 www.mathsrevision.com so 3√125 = 5 Compiled by Mr. Lafferty Maths Dept.**NON - CALCULATOR**6 8 1 1 7 4 3 Mr. Lafferty@mathsrevision.com**Length of a Square**Area of a square is 400cm2 l cm A = l x l = l2 Example : Find the length of the square. www.mathsrevision.com l = √A = √400 l = 20 cm Compiled by Mr. Lafferty Maths Dept.**Special button**on the calculator Compiled by Mr. Lafferty Maths Dept.**Powers & Roots**Now try TJ4+ Ex 10.2 Ch10 (page 65) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.**Starter Questions**X www.mathsrevision.com www.mathsrevision.com**Scientific Notation**Learning Intention Success Criteria • To understand and use the quick method to put large numbers into scientific notation and vice versa. • To show a very quick way of putting a large number in scientific notation very quickly and vice versa. www.mathsrevision.com www.mathsrevision.com**Very large numbers!**BOOM! How far? www.mathsrevision.com 92 000 000 miles**Happy 70th Birthday!**How many seconds in 70 years? Very large numbers! SPLAT! www.mathsrevision.com 70 years = 2 200 000 000 seconds!**Very large numbers!**Dinosaurs roamed the earth 228 million years ago www.mathsrevision.com**The point moved 6 places**Add the decimal point Between 1 and 10 Conversion Move the point to get a number between 1 and 10 = 8 x 10 6 . 8 000 000 . www.mathsrevision.com**The point moved 7 places**Add the decimal point Between 1 and 10 A short cut Move the point to get a number between 1 and 10 = 9.2 x 10 7 . 92 000 000 . www.mathsrevision.com 9.2 x 10 7 miles**The point moved 9 places**Add the decimal point Between 1 and 10 A short cut Move the point to get a number between 1 and 10 = 2.2 x 10 9 . 2 200 000 000 . Happy Birthday: Seconds old! = 2.2 x 10 9**The point moved 8 places**Add the decimal point Between 1 and 10 A short cut Move the point to get a number between 1 and 10 = 2.28 x 10 8 228 000 000 . . www.mathsrevision.com = 2.28 x 10 8 years old**Standard Form**= 3 x 10 4 (1) 30 000 (2) 700 000 (3) 5 300 (4) 470 000 (5) 9 500 000 (6) 18 300 000 (7) 329 000 (8) 2 560 000 (9) 12 000 000 000 (10) 9 990 000 = 7 x 10 5 = 5.3 x 10 3 = 4.7 x 10 5 = 9.5 x 10 6 = 18.3 x 10 7 www.mathsrevision.com = 3.29 x 10 5 = 2.56 x 10 6 = 1.2 x 10 10 = 9.99 x 10 6**The point moves 7 places**Zeros after the point aren’t needed. Changing back 8.6 0000000 . =86 000 000 8.6 x 10 7 = www.mathsrevision.com Hint: Add 7 zeros, although you probably won’t need them all.**The point moves 5 places**Zeros after the point aren’t needed. Changing back 3.46 x 10 5 = 3.46 00000 . =346 000 www.mathsrevision.com Hint: Add 5 zeros, although you probably won’t need them all.**Changing back**(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) = 600 000 = 8 000 = 650 000 = 120 000 000 = 3 710 000 www.mathsrevision.com = 33 000 = 7 910 000 = 55 500 000 = 10 500 = 3 033 000 000**Powers & Roots**Now try TJ4+ Ex 10.3 Ch10 (page 65) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.**Starter Questions**www.mathsrevision.com www.mathsrevision.com**Scientific Notation**Learning Intention Success Criteria • Know how to change very small numbers into scientific notation and how to do the reverse process. • To understand how to change very small numbers into Scientific Notation and vice versa. www.mathsrevision.com www.mathsrevision.com**Very small numbers!**www.mathsrevision.com How wide is an atom? 0.000 000 000 1 metres wide!**The point moved 10 places. Negative sign for small numbers.**Standard Form for small numbers Move the point to get a number between 1 and 10 = 1 x 10 -10 . 0.000 000 000 1 www.mathsrevision.com Scale of SMALL Numbers**The point moved 7 places. Negative sign for small numbers.**Standard Form for small numbers Move the point to get a number between 1 and 10 = 7.6 x 10 -7 . 0.000 000 76 www.mathsrevision.com**The point moved 6 places. Negative sign for small numbers.**Standard Form for small numbers Move the point to get a number between 1 and 10 = 1.93 x 10 -6 0.000 001 93 . www.mathsrevision.com**Standard Form**for small numbers (1) 0. 000 3 (2) 0.000 07 (3) 0.000 45 (4) 0.003 4 (5) 0. 000 724 (6) 0.000 000 494 (7) 0.000 095 (8) 0.000 000 098 (9) 0.000 1 03 (10) 0.000 000 000 66 www.mathsrevision.com**-3 so remember to move point left for small numbers.**Changing back small numbers Hint: Add 3 zeros to the left of the number. 2 x 10 -3 = 2 000 . . www.mathsrevision.com = 0.002**-5 so remember to move point left for small numbers.**Changing back small numbers Hint: Add 5 zeros to the left of the number. . 8.6 x 10 -5 = 8 6 00000 . www.mathsrevision.com = 0.000086**-6 so remember to move point left for small numbers.**Changing back small numbers Hint: Add 6 zeros to the left of the number. 5.16 x 10 -6 = . 5 16 000000 . www.mathsrevision.com = 0.000 00516**Changing back small numbers**(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) = 0.000 5 = 0.000 09 = 0.005 8 = 0.000 000 62 = 0.000 006 45 www.mathsrevision.com = 0.000 000 53 = 0.000 009 17 = 0.002 12 = 0.000 020 3 = 0.000 000 006 032**Powers & Roots**Now try TJ4+ Ex 10.4 Ch10 (page 68) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.**Scientific Notation**Interactive Calculator Lesson www.mathsrevision.com www.mathsrevision.com**Powers & Roots**Now try TJ4+ Ex 10.5 Ch10 (page 70) www.mathsrevision.com Created by Mr. Lafferty Maths Dept.