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Learn about square and triangular numbers, square roots, cubes, cube roots, and index notation in mathematics. Explore the laws of powers and indices, including multiplication and division rules. Practice calculating powers and grasp the concept of negative and zero indices.

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  1. N4 Powers and Roots Contents N4.1 Square and triangular numbers N4.2 Square roots N4.4 Powers N4.3 Cubes and cube roots

  2. We use index notation to show repeated multiplication by the same number. Index notation For example, we can use index notation to write 2 × 2 × 2 × 2 × 2 as Index or power 25 base This number is read as ‘two to the power of five’. 25 = 2 × 2 × 2 × 2 × 2 = 32

  3. Evaluate the following: Index notation When we raise a negative number to an odd power the answer is negative. 62 = 6 × 6 = 36 34 = 3 × 3 × 3 × 3 = 81 (–5)3 = –5 × –5 × –5 = –125 When we raise a negative number to an even power the answer is positive. 27 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128 (–1)5 = –1 × –1 × –1 × –1 × –1 = –1 (–4)4 = –4× –4 × –4 × –4 = 64

  4. 7 4 = xy Calculating powers We can use the xykey on a calculator to find powers. For example, to calculate the value of 74 we key in: The calculator shows this as 2401. 74 = 7 × 7 × 7 × 7 = 2401

  5. When we multiply two numbers written in index form and with the same base we can see an interesting result. The first index law For example, 34 × 32 = (3 × 3 × 3 × 3) × (3 × 3) = 3 × 3 × 3 × 3 × 3 × 3 = 36 = 3(4 + 2) 73 × 75 = (7 × 7 × 7) × (7 × 7 × 7 × 7 × 7) = 7 × 7 × 7 × 7 × 7 × 7 × 7 × 7 = 78 = 7(3 + 5) What do you notice? When we multiply two numbers with the same base the indices are added.

  6. 4 × 4 × 4 × 4 × 4 = 4 × 4 5 × 5 × 5 × 5 × 5 × 5 = 5 × 5 × 5 × 5 When we divide two numbers written in index form and with the same base we can see another interesting result. The second index law For example, 45 ÷ 42 = 4 × 4 × 4 = 43 = 4(5 – 2) 56 ÷ 54 = 5 × 5 = 52 = 5(6 – 4) What do you notice? When we divide two numbers with the same base the indices are subtracted.

  7. Look at the following division: Zero indices 64 ÷ 64 = 1 Using the second index law 64 ÷ 64 = 6(4 – 4) = 60 That means that 60 = 1 In fact, any number raised to the power of 0 is equal to 1. For example, 100 = 1 3.4520 = 1 723 538 5920 = 1

  8. = = 1 3 × 3 1 1 1 1 3 × 3 × 3 × 3 3 × 3 32 32 74 53 1 6 Look at the following division: Negative indices 32 ÷ 34 = Using the second index law 32 ÷ 34 = 3(2 – 4) = 3–2 That means that 3–2 = Similarly, 6–1 = 7–4 = and 5–3 =

  9. 1 an 1 a–1 = a–n= a We can write all of these results algebraically. Using algebra am × an = a(m + n) am÷an=a(m – n) a0 = 1

  10. Using index laws

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