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GCSE Maths
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What you will learn today (something you did not know yesterday) By the end of this session I will be able to⦠Distinguish the different roles played by letter symbols in algebra, using the correct notation. Use vocabulary relating to index numbers Know two special rules relating to index numbers Simplify expressions by applying the first three laws of indices Build confidence in the group and explore maths topics.
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Power Index π π Base
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π π The powers of 2: The powers of π₯: 2 1 =2 π₯ 1 =π₯ 2 2 =2Γ2 π₯ 2 =π₯Γπ₯
2 1 = π₯ 1 =π₯ 2 2 =2Γ π₯ 2 =π₯Γπ₯ 2 3 =2Γ2Γ π₯ 3 =π₯Γπ₯Γπ₯ 2 4 =2Γ2Γ2Γ2 π₯ 4 =π₯Γπ₯Γπ₯Γπ₯ 2 5 =2Γ2Γ2Γ2Γ2 π₯ 5 =π₯Γπ₯Γπ₯Γπ₯Γπ₯ 2 6 =2Γ2Γ2Γ2Γ2Γ2 π₯ 6 =π₯Γπ₯ Γπ₯ Γπ₯ Γπ₯ Γπ₯ π π Base Index
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Copy and complete the table.
Task Copy and complete the table. Use a calculator to help you obtain your answers. We say We write We work out Answer 2 to the power of 4 2 4 2Γ2Γ2Γ2 3 to the power of 4 3Γ3Γ3Γ3 4 4 256 5 to the power of 2 6 5 7776 8Γ8Γ8Γ8Γ8 9Γ9Γ9 3 9 10 to the power of 2 2 to the power of 10 Essential Mathematics (Rayner et al 1998)
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Answers We say We write We work out Answer
2 to the power of 4 2 4 2Γ2Γ2Γ2 16 3 to the power of 4 3 4 3Γ3Γ3Γ3 81 4 to the power of 4 4 4 4Γ4Γ4Γ4 256 5 to the power of 2 5 2 5Γ5 25 6 to the power of 5 6 5 6Γ6Γ6Γ6Γ6 7 776 8 to the power of 5 8 5 8Γ8Γ8Γ8Γ8 32 768 9 to the power of 3 9 3 9Γ9Γ9 729 3 to the power of 9 3 9 3Γ3Γ3Γ3Γ3Γ3Γ3Γ3Γ3 19 683 10 to the power of 2 10 2 10Γ10 100 2 to the power of 10 2 10 2Γ2Γ2Γ2Γ2Γ2Γ2Γ2Γ2Γ2 1024 Essential Mathematics (Rayner et al 1998)
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The same applies to negative numbers, fractions and algebra.
What does (β4) 5 look like in expanded form? β4Γβ4Γβ4Γβ4Γβ4 What does 1 4 Γ 1 4 Γ 1 4 Γ 1 4 look like in index form? What does (3π) 2 mean? 3πΓ3π
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Copy and complete the table.
Task Copy and complete the table. Index form Expanded form 3 2 π₯ 6 7 5 π‘ 7 100 3 (4π) 3 8Γ8 π¦Γπ¦Γπ¦Γπ¦Γπ¦ 1Γ1Γ1Γ1 3π₯Γ3π₯ (β3) 6 (9π₯π¦) 1 7ππΓ7ππΓ7ππ Jo
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Answers Index form Expanded form 3 2 3Γ3 π₯ 6 π₯Γπ₯Γπ₯Γπ₯Γπ₯Γπ₯ 7 5 7Γ7Γ7Γ7Γ7
π‘ 7 π‘Γπ‘Γπ‘Γπ‘Γπ‘Γπ‘Γπ‘ 100 3 100Γ100Γ100 (4π) 3 4πΓ4πΓ4π 8 2 8Γ8 π¦ 5 π¦Γπ¦Γπ¦Γπ¦Γπ¦ 1 4 1Γ1Γ1Γ1 (3π₯) 2 3π₯Γ3π₯ (β3) 6 β3Γβ3Γβ3Γβ3Γβ3Γβ3 (9π₯π¦) 1 9π₯π¦ 2 3 Γ 2 3 Γ 2 3 Γ 2 3 Γ 2 3 Γ 2 3 Γ 2 3 (7ππ) 3 7ππΓ7ππΓ7ππ Jo
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Multiplying with same base number
Example 1β¦ Simplify 83 x 84 = 8x8x8 x 8x8x8x8 = 87 Example 2β¦ Simplify 52 x 54 = 5x5 x 5x5x5x5 = 56 This only works for numbers with the same base number!
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Multiplication Law π π Γ π π =πΓπΓπΓπΓπΓπΓπ = π π π π Γ π π = π π+π βThe base is the same and weβre multiplying the terms, so we add the indicesβ
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Dividing with same base number
Example 1β¦ Simplify 25 Γ· 22 2x2x2x2x2 2x2 23 Example 2β¦ Simplify 35 Γ· 32 3x3x3x3x3 3x3 33 This only works for numbers with the same base number!
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Division Law π π π π = πΓπΓπΓπΓπ πΓπΓπ = π π π π Γ· π π = π πβπ = π πβπ βThe base is the same and weβre dividing the terms, so we subtract the indicesβ
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This only works for numbers with the same base number!
Brackets Example 1β¦ Simplify (35)3 35 x 35 x 35 315 Example 2β¦ Simplify (a3)3 a3 x a3 x a3 a9 This only works for numbers with the same base number!
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βwe multiply the indicesβ
Power Law π π π = π π Γ π π Γ π π = π ππ βwe multiply the indicesβ π π π = π ππ
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Quick Questions 34 x 34 54 x 56 74 Γ· 72 n7 x n9 e16 Γ· e8 (53)2
35 x 3-2 Careful⦠(2-3)-2 Careful 38 510 72 n16 e8 56 33 26
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Special rules of indices
There are some rules (for combining expressions involving indices) Special rule 1: Any number to the power of 1 is the same as the original number (-8)ΒΉ = -8 32ΒΉ = 32 44ΒΉ = 44 5ΒΉ = 5 12ΒΉ = 12 0.4ΒΉ = 0.4 1,000,000ΒΉ = 1,000,000
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Special rules of indices
There are some rules (or laws) for combining expressions involving indices Special rule 2: Any number to the power of 0 is equal to 1 6β° = 1 19β° = 1 0.61β° = 1 35β° = 1 41β° = 1 (-13)β° = 1 1,000,000β° = 1
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Applying skills Apply a technique you feel most comfortable with to answer questions. To get good at maths you must do maths!!
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Back to targets Can you now use probability vocabulary, write probabilities as fractions, decimals or percentages?
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Example: 2 3 Γ 3 2 =1 Task β Find the reciprocals of the following numbers: β π π βπ βπ β π π β π π βπ
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Negative Index ππ π =ππ πππ ππ π =π πππ ππ π =πππ ππ π =ππ ππ π =π ππ βπ = π ππ ππ βπ = π πππ π π =ππ π π =π π π =π π π =π π π =π π βπ = π π π βπ = π π Γ·ππ Γ·π Γ·π Γ·ππ Γ·π Γ·ππ Γ·π Γ·ππ Γ·π Γ·ππ Γ·π Γ·ππ
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A negative index denotes the powerβs reciprocal
= = 1 9 3 β2 = 1 π π π βπ A negative index denotes the powerβs reciprocal
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Negative Indices 1. 3 β1 2. 3 β2 3. 3 β3 β3 β2 β1 β1 β2 = π π = π π = π ππ =ππ =π =π = π π = π π 9. π β1 10. π β2 11. π β3 π β3 π β2 π β1 π π β1 π π β2 = π π = π π π = π π π = π π = π π =π = π π = π π π π
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