Index+Notation

​ Worksheet

= = =Use index notation to write squares such as 2², 3², 4².= =Recognise: • squares of numbers= = = = = = = =Use vocabulary from previous years and extend to:= =// index, indices, index notation, index law… //= = = =Use index notation for small integer powers.= =For example:= = = 144 = 38486 = = =Know that // x //0= 1, for all values of // x //. = = = =Know that: 10 – 1 = 1 ⁄ 10 = 0.1 10 – 2 = 1 ⁄ 100 = 0.01 = = = =Know how to use the // x //// y //key on a calculator to calculate= =powers.= = = = Use ICT to estimate square roots or cube roots to the = = required number of decimal places. For example:= = • Estimate the solution of // x // = 70. = =The positive value of // x //lies between 8 and 9, since 8 2 = 64 and 9 2 = 81. = = = =Try numbers from 8.1 to 8.9 to find a first= =approximation lying between 8.3 and 8.4.= =Next try numbers from 8.30 to 8.40.= = = =Investigate problems such as:= = • Estimate the cube root of 20. • The outside of a cube made from smaller cubes is painted blue.= = = =How many small cubes have 0, 1, 2 or 3 faces painted blue?= =Investigate. • // Three integers, each less than 100, fit the equation //= =// a //// 2 //// + b //// 2 // = c// 2 ////. //= = = =// What could the integers be? //= = = = = = = = Recognise that: = = • = =indices are added when multiplying, e.g.= = = =4 3 × 4 2 = (4 × 4 × 4) × (4 × 4) = = = 4 × 4 × 4 × 4 × 4 = = = 4 5 = 4 (3 + 2) = = = = = = • indices are subtracted when dividing, e.g.= = = =4 5 ÷ 4 2 = (4 × 4 × 4 × 4 × 4) ÷ (4 × 4) = = = 4 × 4 × 4 = = = 4 3 = 4 (5 – 2) = = • 4 2 ÷ 4 5 = 4 (2 – 5) = 4 – 3 = = •  7 5 ÷ 7 5 = 7 0 = 1 = =Generalise to algebra. Apply simple instances of the= =index laws (small integral powers), as in: • // n // 2 × // n // 3 = // n // 2 + 3 = // n // 5 • // p // 3 ÷ // p // 2 = // p // 3 – 2 = // p //=

3 2 means ‘3 squared’, or 3 x 3. The small 2 is an index number, or power. It tells us how many times we should multiply 3 by itself. Similarly 7 2 means ‘7 squared’, or 7 x 7. And 10 2 means ‘10 squared’, or 10 x 10. So, 1 2 =1 x 1= 1 2 2 =2 x 2= 4 3 2 =3 x 3= 9 Etc. 1, 4, 9, 16, 25… are known as square numbers. The opposite of a square number is a square root. We use the symbol to mean square root. So we can say that = 2 and = 5. However, this is not the whole story, because -2 x -2 is also 4, and -5 x -5 is also 25. So, in fact, = 2 or -2. And = 5 or -5. Remember that every positive number has two square roots. 2 x 2 x 2 means ‘2 cubed’, and is written as 2 3. 1 3 =1 x 1 x 1= 1 2 3 =2 x 2 x 2= 8 3 3 =3 x 3 x 3= 27 Etc 1, 8, 27, 64, 125… are known as cube numbers. The opposite of a cube number is a cube root. We use the symbol to mean cube root. So is 2 and is 3. Each number only has one cube root.
 * Squaring a number **
 * Square roots **
 * Cubing a number **
 * Cube roots **

Index notation is used to represent powers, for example a 2 means a × a and here the index is 2 b 3 means b × b × b and here the index is 3 c 4 means c × c × c × c and here the index is 4 etc. When there is a number in front of the variable: 4d 2 means 4 × d × d. 2e 3 means 2 × e × e × e
 * Index notation **

So it follows that: p 3 × p 7 =p= 10, and s 5 ÷ s 3 = = s 2 For the expression: 4s 3 x 3s 2 The numbers in front of the variables follow the usual rules of multiplication and division, but index numbers follow the rules of indices. So we multiply 4 and 3 and add 3 and 2 4s 3 × 3s 2 = 12s 5 Question What is 3c 2 × 5c 4 ?
 * Index laws **
 * Multiplying and dividing **
 * When multiplying you add the indices, and when dividing you subtract the indices. **

Answer Take care when multiplying and dividing expressions such as y × y 4 or z 3 ÷ z. y is the same as y 1, so y × y 4 = y 5. z is the same as z 1, so z 3 ÷ z = z 2.
 * To work it out: **
 * Add the indices:
 * 2 + 4 = 6
 * Multiply the numbers in front of the variable:
 * 3 x 2
 * **Answer: **
 * 3c 2  × 5c 4  = 15c <span style="display: none; font-family: Verdana; font-size: 9pt; mso-bidi-font-family: Arial; mso-hide: all;">6 <span style="display: none; font-family: Verdana; font-size: 10.5pt; mso-bidi-font-family: Arial; mso-hide: all;">

3, 4 and 20 are all like terms (because they are all numbers). a, 3a and 200a are all like terms (because they are all multiples of a). a 2, 10a 2 and -2a 2 are all like terms (because they are all multiples of a 2 ) You cannot simplify an expression like 4p + p 2 because 4p and p 2 are not like terms. But you can simplify 3r 2 + 5r 2 + r 2. 3r 2 + 5r 2 + r 2 tells us that we have ‘three lots of r 2 ’ + ‘five lots of r 2 ’ + ‘one lot of r 2 ’ - so in total ‘nine lots of r 2 ’, or 9r 2. So, 3r 2 + 5r 2 + r 2 = 9r 2 Question What is s 2 + 8s 2 - 2s 2 ?
 * Adding and subtracting **
 * You can only add and subtract ‘like terms’. **

<span style="color: #333333; display: none; font-family: Verdana; font-size: 10.5pt; mso-bidi-font-family: Arial; mso-hide: all;">Answer Remember that 1 + 8 - 2 = 7, so s <span style="color: #333333; display: none; font-family: Verdana; font-size: 8.5pt; mso-bidi-font-family: Arial; mso-hide: all;">2 <span style="color: #333333; display: none; font-family: Verdana; font-size: 10pt; mso-bidi-font-family: Arial; mso-hide: all;"> + 8s <span style="color: #333333; display: none; font-family: Verdana; font-size: 8.5pt; mso-bidi-font-family: Arial; mso-hide: all;">2 <span style="color: #333333; display: none; font-family: Verdana; font-size: 10pt; mso-bidi-font-family: Arial; mso-hide: all;"> - 2s <span style="color: #333333; display: none; font-family: Verdana; font-size: 8.5pt; mso-bidi-font-family: Arial; mso-hide: all;">2 <span style="color: #333333; display: none; font-family: Verdana; font-size: 10pt; mso-bidi-font-family: Arial; mso-hide: all;"> = 7s <span style="color: #333333; display: none; font-family: Verdana; font-size: 8.5pt; mso-bidi-font-family: Arial; mso-hide: all;">2 <span style="color: #333333; display: none; font-family: Verdana; font-size: 10pt; mso-bidi-font-family: Arial; mso-hide: all;"> 3p 2 + 2p + 4 - 2p 2 + 5 = 3p 2 - 2p 2 + 2p + 4 + 5 = p 2 + 2p + 9
 * <span style="color: #333333; display: none; font-family: Verdana; font-size: 10pt; mso-bidi-font-family: Arial; mso-hide: all;">Answer: **<span style="color: #333333; display: none; font-family: Verdana; font-size: 10pt; mso-bidi-font-family: Arial; mso-hide: all;"> 7s <span style="color: #333333; display: none; font-family: Verdana; font-size: 8.5pt; mso-bidi-font-family: Arial; mso-hide: all;">2 <span style="color: #333333; display: none; font-family: Verdana; font-size: 10pt; mso-bidi-font-family: Arial; mso-hide: all;">
 * Remember that if we have a mix of terms we must gather like terms before we simplify. **
 * Example **

You might be asked to substitue a number into an expression. For example, what is the value of 4p 3 when p = 2? We know that 4p 3 means 4 × p × p × p, so when p = 2 we substitute this into the expression: 4 × 2 × 2 × 2 (or 4 × 2 3 ) = 32 Question What is the value of 4y 2 - y, when y = 3?
 * <span style="color: #333333; font-family: Verdana; font-size: 11.5pt; mso-bidi-font-family: Arial; mso-font-kerning: 18.0pt;">Substitution **

**__ Using a calculator __** A calculator can be used to work out one number to the power of another. The index button is usually marked **// xy //** or **// yx //**. Sometimes you need to press ** SHIFT ** or ** 2ndF ** to use this button For example, to calculate **54**, you may need to press

You should find out which buttons you need to use on your calculator. Make sure that you get the correct answer of **625** for the calculation above.

**__ TEST __** <span style="color: #333333; display: block; font-family: Verdana; msobidifontfamily: Arial; text-align: center; text-decoration: none;"> s x s
 * __ Name __**** Date **
 * 1. s² means: **

s + s

2s

2. 2w² means: ** 2w x 2w

2 x w x w

4w

3. y **** 7 **** x y **** 2 **** is the same as: ** y 5

y 9

y 14

4. y **** 10 **** ÷ y **** 2 **** is the same as: ** y 5

y 8

y 20

5. Simplify 4w **** 6 **** x 2w **** 2 **

6w 8

8w 12

8w 8


 * 6. Simplify 14h **** 8 **** ÷ 2h **** 2 **

7h 4

7h 6

12h 6

7. Simplify p x 2p **** 2 **** x 5p **** 3 **

7p 5

10p 5

10p 6

8. What's the value of 4k **** 2 **** when k = 5? ** 20

100

400

9. What's the value of 2j **** 2 **** + j when j = 3? ** 21

39

400

10. What's the value of 2p **** 2 **** + p **** 3 **** - p when p = 2? ** 14

22

40

= = = 1. Find some triples of whole numbers // a // ,  // b //  and  // c //   such that   //a//   2 +   //b//   2 +   //c//   2     is a multiple of 4. Is it necessarily the case that // a // ,  // b //  and  // c //  must all be even? If so, can you explain why? = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = 2. What can you say about the values of n that make = = 7 n + 3 n a multiple of 10? =
 * Are there other pairs of integers between 1 and 10 which have similar properties? **