International Mathematics 0607 Chapter Notes
What this chapter covers
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To square a number, multiply it by itself: 7² = 7 × 7 = 49. A square root undoes a square: √49 = 7 because 7 × 7 = 49.
Learn the squares and their roots for 1 to 15, from 1² = 1 to 15² = 225. The root sign never means "divide by 2".
Worked example 11 mark
Write down the value of √(196).
√(196) = 14, since 14 × 14 = 196 [1]
To cube a number, multiply three copies of it: 4³ = 4 × 4 × 4 = 64. A cube root undoes a cube: √[3]64 = 4.
Learn the cubes and cube roots of 1, 2, 3, 4, 5 and 10: 1, 8, 27, 64, 125 and 1000. Without a calculator, the cube root in a question is always one of these small, easy numbers.
Examiner insight
Worked example 12 marks
Work out.(a) √[3]2197(b) (1.5² - 0.25)/0.4
(a) enter √[3]2197 in the calculator and read off 13 [1]
(b) (2.25 - 0.25)/0.4 = 2/0.4 = 5 [1]
The sign of a power depends on where the brackets are. In (-6)² the whole of -6 is squared, so the answer is positive.
In -6² only the 6 is squared, then the minus sign is applied, so the answer is negative. A negative number squared is always positive; a negative number cubed is negative.
(-6)² = (-6) × (-6) = 36
-6² = -(6 × 6) = -36
Examiner insight
Worked example 11 mark
Complete the statement using one of <, > or =. -6² ... (-6)²
-6² = -36 and (-6)² = 36, so -36 < 36: the sign is < [1]
A number to the power 0 is 1: 1.5⁰ = 1.
For a power of a power, multiply the indices: (3²)⁴ = 3^(2 × 4) = 3⁸. Then work out the number, because "work out" asks for a value, not a power.
a⁰ = 1
(a^(m))ⁿ = a^(mn)
Examiner insight
Worked example 12 marks
Work out.(b) 1.5⁰(c) (3²)⁴
(b) any number to the power 0 is 1 [1]
(c) (3²)⁴ = 3⁸ = 6561 [1]
Use the power key for any index: enter the base, the power key, then the index. Use the cube-root template for √[3] and the root template for √.
A reciprocal such as 1/3.5² is worked out in two stages: square first, then divide. Round only at the end, to the accuracy asked: 3 decimal places is not 3 significant figures.
1/a² = 1 ÷ (a × a)
Examiner insight
Worked example 13 marks
Work out.(a) √[3]3375(b) 1/3.5². Give your answer correct to 3 decimal places.
(a) enter √[3]3375 and read off 15 [1]
(b) 3.5² = 12.25, so 1/12.25 = 0.0816… [1]
= 0.082 to 3 decimal places [1]
Recap
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