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A perfect square is a number that can be expressed as the product of an integer multiplied by itself. For example, numbers like 1, 4, 9, 16, 25, and 64 are perfect squares. Here’s why:

• 1 is \( 1 \times 1 \)
• 4 is \( 2 \times 2 \)
• 9 is \( 3 \times 3 \) and so on.

64 is a perfect square because \( 8 \times 8 = 64 \). Knowing how to identify perfect squares is very helpful when working with square roots because it allows you to quickly find the square root of a number.
In our exercise, the number 64 is easily identifiable as a perfect square, which makes finding the square root straightforward.
Recognizing perfect squares can save time and help simplify your calculations in math.

Multiplication is one of the fundamental operations in arithmetic. It involves finding the total number of items in groups of the same size. For numbers, it can be thought of as repeated addition. For example, multiplying 3 by 4 \( (3 \times 4) \) is the same as adding three groups of four, or \( 4 + 4 + 4 \).

In the context of perfect squares and square roots, multiplication plays a key role. If you can multiply a number by itself to reach another number, then you've identified a perfect square. In our earlier example:

• \( 8 \times 8 = 64 \)

This operation helped us quickly solve the square root of 64 because we knew beforehand that 8 is the integer involved.

Being comfortable with multiplication will also help you verify your results when checking if a number is a perfect square. It is a crucial skill in mastering more complex math concepts.

Negative numbers are any numbers less than zero, and they are represented with a minus sign (e.g., -1, -2, -3). They are often used to express values such as debts, temperatures below freezing, or depths below sea level.

In math problems involving square roots, negative signs can sometimes cause confusion. It’s important to remember that while you can have the negative square root of a number, like \(-\sqrt{64}\), a square root itself is always considered the positive value by default.

When finding \(-\sqrt{64}\), you first find the positive square root, which we've identified as 8, and then you apply the negative sign: \(-8\). Remember:

• A negative sign in front of a square root signifies a negative value of that root.
• The square root of any positive number will have both a positive and a negative root.

Understanding negative numbers can help you interpret not only square root operations, but also gain a strong grasp in number line concepts and real-world applications.