CIE IGCSE Mathematics (0580) Powers and roots Study Notes - New Syllabus
CIE IGCSE Mathematics (0580) Powers and roots Study Notes
LEARNING OBJECTIVE
- Powers and Roots
Key Concepts:
- squares & square roots
- cubes & cube roots
- other powers and roots of numbers
Powers and Roots
Powers and Roots
Squares
Squaring a number means multiplying it by itself. The square of a number \( a \) is written as \( a^2 \).
Examples: \( 2^2 = 4, \quad 5^2 = 25, \quad (-3)^2 = 9 \)
Note: Squaring a negative number gives a positive result.
Square Roots
The square root of a number \( x \) is the number that gives \( x \) when squared. It is written as \( \sqrt{x} \). Only non-negative numbers have real square roots in IGCSE mathematics.
Examples: \( \sqrt{25} = 5, \quad \sqrt{49} = 7, \quad \sqrt{0} = 0 \)
Note: \( \sqrt{a^2} = |a| \) because square root always gives the non-negative (principal) value.
| Number | Square | Square Root |
|---|---|---|
| 1 | 1 | \( \sqrt{1} = 1 \) |
| 2 | 4 | \( \sqrt{4} = 2 \) |
| 3 | 9 | \( \sqrt{9} = 3 \) |
| 4 | 16 | \( \sqrt{16} = 4 \) |
| 5 | 25 | \( \sqrt{25} = 5 \) |
| 6 | 36 | \( \sqrt{36} = 6 \) |
| 7 | 49 | \( \sqrt{49} = 7 \) |
| 8 | 64 | \( \sqrt{64} = 8 \) |
| 9 | 81 | \( \sqrt{81} = 9 \) |
| 10 | 100 | \( \sqrt{100} = 10 \) |
| 11 | 121 | \( \sqrt{121} = 11 \) |
| 12 | 144 | \( \sqrt{144} = 12 \) |
| 13 | 169 | \( \sqrt{169} = 13 \) |
| 14 | 196 | \( \sqrt{196} = 14 \) |
| 15 | 225 | \( \sqrt{225} = 15 \) |
Cubes
Cubing a number means multiplying it by itself twice. The cube of a number \( a \) is written as \( a^3 \).
Examples: \( 2^3 = 8, \quad 3^3 = 27, \quad (-4)^3 = -64 \)
Note: Cubes can be positive or negative depending on the sign of the number.
Cube Roots
The cube root of a number \( x \), written as \( \sqrt[3]{x} \), is the number which, when cubed, gives \( x \). Cube roots can be positive or negative.
Examples: \( \sqrt[3]{8} = 2, \quad \sqrt[3]{-27} = -3, \quad \sqrt[3]{0} = 0 \)
| Number | Cube | Cube Root |
|---|---|---|
| 1 | 1 | \( \sqrt[3]{1} = 1 \) |
| 2 | 8 | \( \sqrt[3]{8} = 2 \) |
| 3 | 27 | \( \sqrt[3]{27} = 3 \) |
| 4 | 64 | \( \sqrt[3]{64} = 4 \) |
| 5 | 125 | \( \sqrt[3]{125} = 5 \) |
| 10 | 1000 | \( \sqrt[3]{1000} = 10 \) |
Other Powers
Powers beyond squares and cubes include expressions like \( a^4, \; a^5 \), and so on. In general, \( a^n \) means multiplying \( a \) by itself \( n \) times.
Examples: \( 3^4 = 81, \quad 2^5 = 32 \)
Other Roots
Roots higher than square or cube include the fourth root \( \sqrt[4]{x} \), fifth root \( \sqrt[5]{x} \), etc. These are the inverse operations of higher powers.
Examples: \( \sqrt[4]{16} = 2, \quad \sqrt[5]{32} = 2 \)
Example:
Calculate: (a) \( 7^2 \) (b) \( \sqrt{121} \) (c) \( \sqrt{(-5)^2} \)
▶️ Answer/Explanation
(a) \( 7^2 = 7 \times 7 = 49 \)
(b) \( \sqrt{121} = 11 \), because \( 11^2 = 121 \)
(c) \( \sqrt{(-5)^2} = \sqrt{25} = 5 \) (always non-negative)
Example:
Calculate: (a) \( (-2)^3 \) (b) \( \sqrt[3]{-64} \)
▶️ Answer/Explanation
(a) \( (-2)^3 = -2 \times -2 \times -2 = -8 \)
(b) \( \sqrt[3]{-64} = -4 \), because \( (-4)^3 = -64 \)
Note: Cube roots of negative numbers are also negative.
Example:
Evaluate: (a) \( 3^4 \) (b) \( \sqrt[4]{81} \) (c) \( 2^5 \)
▶️ Answer/Explanation
(a) \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \)
(b) \( \sqrt[4]{81} = 3 \), since \( 3^4 = 81 \)
(c) \( 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \)