3.1 Squares and Square Roots
1. Squares of Numbers
The square of a number is the product of the number multiplied by itself.
- Notation: The square of $a$ is written as $a^2 = a \times a$.
- Example: $5^2 = 5 \times 5 = 25$.
- Positive & Negative Base: The square of any number (positive or negative) is always positive. For example, $(-4)^2 = (-4) \times (-4) = 16$.
2. Perfect Squares
A perfect square is a whole number produced by multiplying a whole number by itself.
- Examples: $1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144...$
- Testing for Perfect Squares: Perform prime factorisation. If prime factors can be grouped into two identical sets, the number is a perfect square.
3. Square Roots
The square root of a number $n$ is a number which, when multiplied by itself, equals $n$.
- Notation: The square root is denoted by the symbol $\sqrt{n}$.
- Relationship: If $a^2 = b$, then $\sqrt{b} = a$.
- Example: Since $6^2 = 36$, $\sqrt{36} = 6$.
- Key Property: $(\sqrt{a})^2 = a$ and $\sqrt{a} \times \sqrt{a} = a$.
3.2 Cubes and Cube Roots
1. Cubes of Numbers
The cube of a number is the product of the number multiplied by itself twice.
- Notation: The cube of $a$ is written as $a^3 = a \times a \times a$.
- Example: $3^3 = 3 \times 3 \times 3 = 27$.
- Sign Rules:
- The cube of a positive number is positive: $2^3 = 8$.
- The cube of a negative number is negative: $(-2)^3 = (-2) \times (-2) \times (-2) = -8$.
2. Perfect Cubes
A perfect cube is a whole number produced by multiplying a whole number by itself twice.
- Examples: $1, 8, 27, 64, 125, 216, 343, 512, 729, 1000...$
- Testing for Perfect Cubes: Perform prime factorisation. If prime factors can be grouped into three identical sets, the number is a perfect cube.
3. Cube Roots
The cube root of a number $n$ is a number which, when multiplied by itself twice, equals $n$.
- Notation: The cube root is denoted by the symbol $\sqrt[3]{n}$.
- Relationship: If $a^3 = b$, then $\sqrt[3]{b} = a$.
- Negative Numbers: $\sqrt[3]{-a} = -\sqrt[3]{a}$. For example, $\sqrt[3]{-27} = -3$.
- Key Property: $(\sqrt[3]{a})^3 = a$ and $\sqrt[3]{a^3} = a$.
3.3 Combined Operations & Estimation
1. Estimation
To estimate square roots or cube roots of numbers that are not perfect squares or cubes, identify the closest perfect squares or cubes that bound the given number.
- Example: To estimate $\sqrt{20}$, since $16 < 20 < 25$, we have $\sqrt{16} < \sqrt{20} < \sqrt{25}$. Therefore, $4 < \sqrt{20} < 5$, so $\sqrt{20} \approx 4.5$.
2. Combined Operations
When solving expressions involving squares, square roots, cubes, and cube roots, follow the standard order of operations (BODMAS):
- Evaluate roots and powers first (or terms inside brackets).
- Perform multiplication and division from left to right.
- Perform addition and subtraction from left to right.