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GCSE Maths - Laws of Indices
06/08/2023
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GCSE MATHS LAWS OF INDICES Terminology "exponent" or "index" (plural "indices”) 34 = 3 × 3 × 3 × 3 "power" "Base" The exponent tells us how many times the base appears in a product. We say this as "3 to the power of 4" or "3 raised to the power of 4" or "3 to the 4". More on notation 34 Phrase "Powers of 2." You may have heard the 4 referred to as the power. But 'power' refers to the whole expression 34; the 4 is the exponent! "3 raised to the power of 2." Precise meaning Powers where the base is 2: 2¹, 22, 23, 24 32. 'Raised' here means 'turned into a power'. While "power of 2" might suggest the 2 is the 'power', it's really short for "power with an exponent of 2". Understanding powers When the exponent is a positive integer (whole number), it indicates how many times the base is repeated in the multiplication. 5 appears 2 times. 5² = 5 x 5 = 25 23 = 2 × 2 × 2 = 8 34 = 3 × 3 × 3 × 3 = 81 25 = 2 × 2 × 2 × 2 × 2 = 32 13¹ 13 It's possible for the exponent to be fractional, 0 or negative. We'll deal with these later! Warning: Sometimes people incorrectly describe "4³" as "4 multiplied by itself 3 times". This...
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Alternative transcript:
would suggest there are 3 multiplications, but 4 x 4×4 actually only has 2 multiplications! Multiplying powers How would we simplify this? x³ means 3 x's .3 multiplied together. x3 x x2 = xXxXx x5 = In total we had 5 x's multiplied together. 1st law of indices: xa xxb = xa+b хать X X X X i.e. when we multiply two powers, we add the exponents. Dividing Powers How would we simplify this? x5 x3 = xxxxxxxxx xxxxx x² 2 2nd law of indices: x@ : xb = xa-b i.e. when we divide two powers, we subtract the exponents. Remember that we can simplify fractions by dividing the numerator and denominator by the same number (or term). Raising a Power to a Power How would we simplify this? (x³)4 = x³ x x ³ = x12 x³ x x³ x x³ x x³ 3rd law of indices: (xª)b = xab i.e. when we raise a power to a power, we multiply the exponents. Zero and negative indices 3⁰ 3-1 At this point, it doesn't make sense to say "The product of -1 threes". We'll have to use a different approach! Is there a pattern we can see that will help us out? 3³ = 27 329 3¹ = 3 3⁰ = 1 1 -1 3-¹ = ²/3 3-² = 1 1/2 9 ÷ 3 ÷ 3 ÷ 3