Indices – Learning Outcomes Use and apply rules for indices: 𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝+𝑞 𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝−𝑞 𝑎 𝑝 𝑞 = 𝑎 𝑝𝑞 Use the notation 𝑎 1 2 Express rational numbers ≥ 1 in the form 𝑎× 10 𝑛 , where 𝑎 is a decimal and 𝑛 is a natural number. (Use Scientific notation).

Use and Apply Rules for Indices Recall that multiplication represents repeated addition. e.g. 2+2+2+2+2=5×2 Indices are used to show repeated multiplication: e.g. 2×2×2×2×2= 2 5 e.g. write the following numbers using indices: 2 is the “base” 5 is the “power” or “exponent” 3×3×3×3 4×4×4×4×4×4 8×8×8 1×1×1×1 5

Use and Apply Rules for Indices e.g. Write out the following indices as numbers multiplied by themselves: 2 3 7 2 8 5 9 4 4 1 5 10

Use and Apply Rules for Indices Recall that adding and subtracting are opposites, and that multiplying and dividing are opposites. The opposite of an index is a root. e.g. 7 2 is “seven squared”, 7 is “the square root of seven”. e.g. Use a calculator to find the values of the following: 4 9 36 10000 5 2 25 1 2 49 1 2 1 1 2 8 2 1 2

Use and Apply Rules for Indices What happens when numbers in index form are multiplied? e.g. What is 2 5 × 2 3 in index form? e.g. What is 3 2 × 3 4 in index form? In general, what do you do to the power when multiplying numbers in index form? 𝑎 𝑝 × 𝑎 𝑞 = 𝑎 𝑝+𝑞

Use and Apply Rules for Indices 𝑎 𝑝 × 𝑎 𝑞 = 𝑎 𝑝+𝑞 Write the following in index form using the rule for adding powers: 3 5 × 3 3 7 2 × 7 8 2 3 × 2 1 6 2 × 6 8 × 6 3 4 1 × 4 9 × 4 2 𝑦 4 × 𝑦 2 𝑧 3 × 𝑧 10 × 𝑧 3

Use and Apply Rules for Indices 2012 OL P1 Q3 The table shows the values when 2 is raised to certain powers. Complete the table. Maria wins a lottery and is given two options: Option A: €1000 today Option B: €2 today, €4 tomorrow, €8 the next day, doubling every day for 9 days. Power of 2 Expanded Power of 2 Answer 2 1 2 2 2 2×2 4 2 3 2×2×2 2 4 2 5 2 6 2 7 2 8 2 9 Which is the better option?

Use and Apply Rules for Indices What happens when numbers in index form are divided? e.g. What is 2 5 2 3 in index form? e.g. What is 3 2 3 4 in index form? In general, what do you do to the power when dividing numbers in index form? 𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝−𝑞

Use and Apply Rules for Indices 𝑎 𝑝 𝑎 𝑞 = 𝑎 𝑝−𝑞 Write the following in index form using the rule for subtracting powers: 3 5 3 3 7 2 7 8 2 3 2 1 6 2 6 8 𝑦 4 𝑦 2 𝑧 3 𝑧 10 2 5 × 2 4 2 3 6 2 × 6 7 6 11 3 4 × 3 8 3 2 × 3 6 5 2 × 5 4 5 9 × 5 2 𝑦 3 × 𝑦 8 𝑦 5 𝑧 4 × 𝑧 2 𝑧 9 × 𝑧 3 × 𝑧 4

Use and Apply Rules for Indices 2004 OL P1 Q2 Simplify 𝑎 7 × 𝑎 4 𝑎 3 × 𝑎 2 , giving your answer in the form 𝑎 𝑛 , where 𝑛∈ℕ. 2005 OL P1 Q2 Simplify 𝑎 9 × 𝑎 5 𝑎 6 × 𝑎 2 , giving your answer in the form 𝑎 𝑛 , where 𝑛∈ℕ.

Use and Apply Rules for Indices What happens when numbers in index form are raised to a power? e.g. What is 2 5 3 in index form? e.g. What is 3 2 4 in index form? In general, what do you do to the power when raising a number in index form to a power? 𝑎 𝑝 𝑞 = 𝑎 𝑝×𝑞

Use and Apply Rules for Indices 𝑎 𝑝 𝑞 = 𝑎 𝑝×𝑞 Write the following in index form using the rule for multiplying powers: 3 5 3 7 2 8 2 3 1 6 2 8 3 4 1 9 2 𝑦 4 2 𝑧 3 10 3

Use and Apply Rules for Indices 2006 OL P1 Q2 Using a calculator or otherwise, find the exact value of 4 2 3 . 2011 OL P1 Q2 Write 𝑎 3 2 in the form 𝑎 𝑛 , 𝑛∈ℕ. Using your answer from (i) or otherwise, evaluate 5 3 2 . 2011(S) LC OL P1 Q2 Show that 𝑎 𝑎 3 𝑎 4 simplifies to 𝑎

Use Scientific Notation Scientific notation splits up the size of a number from its digits. It is used to show very large and very small numbers. e.g. 1 234 567 891 234 567 is a very long number. Scientific notation takes its first few digits (123…) and makes it a decimal (1.23) It also takes its size (16 digits long) and makes it a power of ten ( 10 15 ) – the power is 1 less than the number of digits. So 1 234 567 891 234 567 is written 1.23× 10 15 in scientific notation, which is shorter.

Use Scientific Notation To enter scientific notation mode on a CASIO, press [SHIFT], [SET UP], [7:Sci], [3] (the last one is how many digits you want). To leave scientific notation mode on a CASIO, press [SHIFT], [SET UP], [8:Norm], [1]. e.g. enter scientific notation mode on your calculator and write the following numbers in scientific notation: 300 4 321 768 53 000 -6 720 000 000 0.2 0.000 000 007 51 1 230 400 2

Use Scientific Notation

Use Scientific Notation 2006 OL P1 Q2 Using a calculator or otherwise, multiply 65.5 by 40 and express your answer in the form 𝑎× 10 𝑛 , where 1≤𝑎<10 and 𝑛∈ℤ. 2007 OL P1 Q2 Using a calculator or otherwise, multiply 54.5 by 60 and express your answer in the form 𝑎× 10 𝑛 , where 1≤𝑎<10 and 𝑛∈ℤ. 2010 OL P1 Q2 Using a calculator or otherwise, divide 1120 by 0.035 and express your answer in the form 𝑎× 10 𝑛 , where 1≤𝑎<10 and 𝑛∈ℤ. 2011(S) LC OL P1 Q2 Express 2 24 the form 𝑎× 10 𝑛 , where 1≤𝑎<10 and 𝑛∈ℤ.