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The concept of the **principal square root** is the main idea when dealing with square roots. It refers to the non-negative root of a number. For a positive real number like 64, its principal square root is the number that, when multiplied by itself, equals 64. This is represented as \( \sqrt{64} = 8 \). The principal square root always gives us the positive value.

It’s important to remember that every positive number actually has two square roots: a positive and a negative one. However, by convention, when we refer to \( \sqrt{64} \), we mean the principal square root, which is 8 in this case.

The principal square root is particularly useful in scenarios where only positive numbers are applicable, such as in measurements or when dealing with lengths in geometry.

The **negative square root** of a number is equally important to understand. Unlike the principal square root, the negative square root refers to the negative value that, when squared, equals the original number. For the number 64, the negative square root is \( -8 \).

Why does this work? Because multiplying \(-8\) by itself, \((-8) \times (-8)\), also results in 64. So, every positive number has two square roots: one positive (the principal) and one negative.

When an exercise specifically asks for the negative square root, it's crucial to remember to include the negative sign. For example, if asked to find the negative square root of 64, the answer would be \( -\sqrt{64} = -8 \).

• Always check what root is being asked for: principal or negative.
• For the negative root, remember the negative sign!

A **perfect square** is a number that is the square of an integer. The number 64 is considered a perfect square because it can be expressed as \( 8^2 \). In simpler terms, if you multiply 8 by itself, you get 64.

Understanding perfect squares is helpful because it makes calculating square roots easier. If you know a number is a perfect square, you can quickly find its root without a calculator. For example, since 64 is a perfect square, its roots are easily identified as 8 and -8, the principal and negative square roots respectively.

Here are some features of perfect squares:

• They come from squaring whole numbers.
• They always have two whole number roots: one positive and one negative.
• Recognizing perfect squares can simplify your math work!