Overview

This cov­ers; Laws of Indices, Frac­tional and Neg­a­tive Pow­ers, Stan­dard Form (build­ing on knowl­edge gained in Com­plex Cal­cu­la­tions and Accu­racy) and Surds. Some of this is quite com­pli­cated (some Grade A and A* ques­tions). I think the trick is to make sure you under­stand the basics thor­oughly before mov­ing on to the more advanced stuff.

Indices and Stan­dard Form Questions

  1. Work out the value of;
  2. Work out the value of;

a) (3³ x 5² x 10)  x  (3–2 x 5–1 x 10²)

  1. Cal­cu­late these expressions:

a) (10–2)–2

b) 9² x 3√27  ÷  4√320

  1. Cal­cu­late the value of:-

a) 811/4

b) 62525

  1. Write the fol­low­ing using index notation:

a) (4√13)4

b) (4√11)2

  1. Cal­cu­late the value of:-

a) 216−2÷3

b) 81−3÷4

  1. Work out the fol­low­ing with answers in stan­dard form:-

a) (3.2 x 10–5) x (8 x 10–3)

b) (3 x 10–7) ÷ (8 x 10–1)

c) (1.6 x 104) + (2.4 x 105)

d) (6.07 x 105) — (1.8 x 103)

  1. Write this expres­sion in the form x — y√8, x and y are integers:-

(√32 — 2)(√72 + 1)

Indices and Stan­dard Form Approach

  1. These look scary but you can just break them down in a few steps. Let’s look at ques­tion 1a):- 

See final answer to parts a) and b) below

  1. Again this is worse than it looks. You can just elim­i­nate the brack­ets and add and sub­tract the pow­ers (remem­ber that x1 = x) so 10 x 10² = 103
  2. a) Mul­ti­ply the indices but remem­ber that a neg­a­tive times a neg­a­tive is a positive

b) In this ques­tion you need to divide the indices using rule:-  x√Wy = W(y/x)

  1. Start with the knowl­edge that X1/y = y√X and for part (b) 0.25 = 1/4, so the same rule can be applied.
  2. Again, use the rule — x√Wy = W(y/x)

Remem­ber that the ques­tion asks for answers in index nota­tion i.e xy

  1. Yet another rule to use: x–y/z =  1/xy/z
  2. Work out the fol­low­ing with answers in stan­dard form:-

Remem­ber to give answers in stan­dard form i.e. Y x 10n (with Y a num­ber between 1 and 10 and n is an integer).

  1. This is most def­i­nitely an A* ques­tion. I got really con­fused and it took me some time to crack (more than you get in an exam!) — please look at the detailed steps to see the approach I took.

Indices and Stan­dard Form Answers

  1. a)

54  x 33 = 625 x 27 =16,875

b) 64 x 10³ = 1,296,000

  1. (3³ x 5² x 10)  x  (3–2 x 5–1 x 10²) = 3 x 5 x 103 = 15,000
  2. a) (10–2)–2 = 104 = 10,000

b) 9² x 3√27  ÷  4√320

= 81 x 3 x 35 = 59,049

  1. a) 811/4 = 4√81 = 3

b) 62525 =  4√625= 5

  1. a) (4√13)4 = 131 = 13

b) (4√11)2 = 112/4 = 115

  1. a) 216−2÷3 = 1/2162/3

= 1/3√2162  = 1/62 = 1/36

  1. b) 81−3÷4 = 1/813/4

=  1/4√813  = 1/33 = 1/27

  1. a) (3.2 x 10–5) x (8 x 10–3)  = 25.6 x 10–8 =2.56 x 10–7

b) (3 x 10–7) ÷ (8 x 10–1) = 3/8 x 10–7/10–1

=0.375 x 10–6  = 3.75 x 10–7

c) (1.6 x 104) + (2.4 x 105) = 16,000 + 240,000 = 256,000 = 2.56 x 105

d) (6.07 x 105) — (1.8 x 103) = 607,000 — 1,800  = 605,200 =6.052 x 105

  1. (√32 — 2)(√72 + 1) = (√4√8 — 2)(√9√8 + 1) = (2√8 — 2)(3√8 + 1)

(When mul­ti­ply­ing out the first part of this sum note that: 2√8  x 3√8 = 2 x √8 x √8 x 3 = 6 x 8 = 48)

There­fore (2√8 — 2)(3√8 + 1) = 48 + 2√8 — 6√8 — 2 =

46 — 4√8