Squares and Square Roots Study Notes

Square of a Number: Obtained by multiplying a whole number by itself ( $a^2 = a \times a$ ).
- Read as " $a$ squared" (e.g., $8^2$ is read as " $8$ squared").
- Example: $8^2 = 8 \times 8 = 64$ , $5^2 = 5 \times 5 = 25$ .
Perfect Square: Any whole number that is the product of a whole number multiplied by itself (e.g., $25$ and $64$ are perfect squares).
Geometric Connection: Finding the square of a number corresponds to determining the area of a square with that side length:
- $\text{Area} = \text{length} \times \text{length}$
- A square with sides of length $8\,\text{cm}$ has an area of $8\,\text{cm} \times 8\,\text{cm} = 64\,\text{cm}^2$ .

Square Root ( $\sqrt{}$ ): The inverse operation of squaring a number; it finds the number that must be multiplied by itself to equal a given value.
- Notation: $\sqrt{b} = a$ implies $a^2 = b$ .
- Example: $\sqrt{9} = 3$ because $3 \times 3 = 9$ .
Geometric Connection: Determining the square root of an area yields the side length of that square:
- A square with an area of $9\,\text{m}^2$ has side lengths of $3\,\text{m}$ .
- A square with an area of $100\,\text{units}^2$ has side lengths of $\sqrt{100} = 10\,\text{units}$ .
Evaluation Methods:
- Multiplication Facts: Recalling known whole-number products (e.g., $10 \times 10 = 100 \implies \sqrt{100} = 10$ ).
- Prime Factorization: Expressing the number as a product of prime factors, grouping them into two identical products, and extracting the base:
- $100 = 2 \times 2 \times 5 \times 5 = (2 \times 5) \times (2 \times 5) = 10^2 \implies \sqrt{100} = 10$

Definition: Whole numbers whose square roots are not whole numbers.
Properties:
- The square root of a non-perfect square is a non-terminating, non-repeating decimal.
- Example: $\sqrt{1000} = 31.622776601\dots$
Approximation: Calculators provide rounded values to approximate the square roots of non-perfect squares.