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When we talk about the square root of a number, we are often referring to the positive square root. This is the number that, when multiplied by itself, results in the original number. For instance, if you have the number 25, you are looking for a number that, squared, equals 25. The positive square root of 25 is 5 because when you multiply 5 by 5, you get 25. It's important to remember that the square root symbol, \( \sqrt{} \), typically refers to the positive square root.

• The positive square root of 36 is 6, since \( 6 \times 6 = 36 \).
• The positive square root of 49 is 7, since \( 7 \times 7 = 49 \).

Keep in mind that positive square roots are always positive numbers.

While the positive square root is most commonly referenced, every positive number actually has a negative square root as well. The negative square root is just the opposite (or negative) of the positive square root but will also square to the original number. Using 25 as an example, if the positive square root is 5, the negative square root is -5. This is because when you multiply \(-5\) by \(-5\), the result is also 25, due to the property of multiplication that a negative number times another negative number gives a positive result. Some more examples include:

• The negative square root of 9 is -3, because \(-3 \times -3 = 9\).
• The negative square root of 64 is -8, because \(-8 \times -8 = 64\).

Using both positive and negative square roots broadens our understanding of numbers, allowing us to consider all possible roots.

Multiplication is a mathematical operation where you combine different numbers. This operation is integral to understanding square roots, as finding a square root involves determining what number must be multiplied by itself to get the original number. For example, in determining the square roots of 25, you check which number multiplied by itself equals 25. Multiplication also follows certain properties that are essential to understanding how negative square roots work:

• Commutative Property: The order of numbers does not change the result, meaning \( a \times b = b \times a \).
• Associative Property: The way numbers are grouped does not affect the result, meaning \( (a \times b) \times c = a \times (b \times c) \).
• Multiplying Negatives: Two negative numbers multiplied together produce a positive result.

Understanding these properties of multiplication helps in grasping how both positive and negative square roots relate to the original number through this operation.