Indices Vocabulary: index square root indices cube base cube root power integer exponent evaluate prime index laws composite zero index divisibility verify factors evaluate square index notation Indices

Main points covered in this unit: Indices Main points covered in this unit: Describe numbers using index form Evaluate numbers expressed using powers Use divisibility tests correctly Write numbers using prime factors Find the highest common factor of numbers Use order of operations Use index laws to simplify expressions

Why use indices? Indices According to legend, chess was invented by Grand Vizier Sissa Ben Dahir, and given to King Shirham of India. The king offered him a reward, and he requested the following: "Just one grain of wheat on the first square of the chessboard. Then put two on the second square, four on the next, then eight, and continue, doubling the number of grains on each successive square, until every square on the chessboard is reached."

Indices Writing Indices

Put the terms in the correct columns: Indices Put the terms in the correct columns: 3 2 3 4 2 3 3 3 3 10 2 4 2 2 10 4 5 3

Put the terms in the correct columns: Indices Put the terms in the correct columns: 3 10 3 4 2 4 10 4 2 2 3 2 5 3 3 3 2 3

Indices Index Laws – short cuts! We can add the indices when multiplying bases that are the same. 𝑎 𝑚 × 𝑎 𝑛 = 𝑎 𝑚+𝑛 Whole numbers in front of the bases can be multiplied as usual. So 3 4 × 3 6 = 3 10 And: 𝑎 𝑚 ÷ 𝑎 𝑛 = 𝑎 𝑚−𝑛 So 3 8 ÷ 3 2 = 3 6 And: ( 𝑎 𝑚 ) 𝑛 = 𝑎 𝑚×𝑛 So ( 3 2 ) 4 = 3 8 What about 𝑎 0 ? 𝑎 0 =1

Indices

Indices

What’s the difference between a factor and a multiple? Indices What’s the difference between a factor and a multiple?

Indices

Indices

Prime Factor Trees Indices The highest common factor (HCF) of two numbers is the biggest number which divides evenly into both of them. Two ways to do this are using lists, and using factor trees. e.g. Find the HCF of 36 and 64: 36: 1,2,3,4,6,9,12,18,36 64: 1,2,4,8,16,32,64 So HCF of 36 and 64 is 4 = 2×2×3×3 = 2 2 × 3 2 64 = 2×2×2×2×2×2 = 2 6 Look for the common factors in the bottom line and multiply them all together – in this case ‘2 x 2’ which also gives 4.

Indices

Indices Create a factor tree for each number, then use them to find the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) for each pair. a) 20 36 b) 60 16 c) 15 35 d) 28 36

Divisibility tests: How do we know if a number has factors? Indices Divisibility tests: How do we know if a number has factors?

Indices

Squares and Cubes Indices A square number can be represented as a square A square root is the side length of the square in the diagram above. So 1 =1; 4 =2; 9 =3; 16 =4; A cube number can be represented as a ‘3D’ cube. A cube root is the side length of the cube 3 1 =1; 3 8 =2; 3 27 =3; 3 64 =4;

Indices

Indices