Squares
Definition
A square of a number means multiplying it by itself.
Notation: n² and read as: “n squared.”
Explanation in words: the number × itself.
Examples:
4² = 4 × 4 = 16
7² = 7 × 7 = 49
Practice Problem
Calculate 6².
Answer: 6 x 6= 36
Square Roots
Definition: the square root is the number which, when multiplied by itself, gives the original number.
Notation: √n and read as: “square root of n.”
Explanation in words: the opposite of squaring.
Examples:
√25 = 5 because 5 × 5 = 25.
√81 = 9 because 9 × 9 = 81.
Important:
Only positive square roots are considered in IGCSE unless stated.
√ symbol always means the principal (positive) root.
Practice Problem
Find √64.
Answer: 8 since 8 × 8 = 64
Cubes
Definition
A cube of a number is that number multiplied by itself twice.
Notation: n³ and read as: “n cubed.”
Explanation in words: n × n × n.
Examples:
2³ = 2 × 2 × 2 = 8
4³ = 4 × 4 × 4 = 64
Practice Problem
Calculate 3³.
Answer: 3 × 3× 3 = 27
Cube Roots
Definition: the cube root is the number which, when cubed, gives the original number.
Notation: ∛n or n^1/3 and read as: “cube root of n.”
Explanation in words: the opposite of cubing.
Examples:
∛27 = 3 because 3 × 3 × 3 = 27.
∛64 = 4 because 4 × 4 × 4 = 64.
Important:
Cube roots can be positive or negative like ∛–8 = –2. (n can be positive or negative since a negative number can be multiplied by itself twice and still give negative whereas in squaring, you cant multiply a negative number by itself and expect a negative output)
Example: cubing: (-3) x (-3) x (-3) = -27 but (-3) x (-3) = 9 (not -9)
Practice Problem
Find ∛125.
Answer: 5
Other Powers
Definition
Powers are numbers raised to any exponent (not just 2 or 3).
Notation: aⁿ and read as: “a to the power of n.”
Explanation in words: multiply a by itself n times.
Examples:
5⁴ = 5 × 5 × 5 × 5 = 625.
2⁵ = 32.
Important:
Exponents tell how many times to multiply.
Powers can be positive, negative, fractional (covered in indices later).
Practice Problem
Calculate 2⁴.
Answer: 2 x 2 x 2 x 2 = 16
NOTES DONE BY FARIDA SABET
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