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When we talk about the square roots of a number, we dive into the world of finding special numbers that when multiplied by themselves, return the original number. For the number 36, one of these special numbers is its positive root.

The positive root of 36 is indeed 6. This can be expressed mathematically as \[\sqrt{36} = 6\]The positive root is the value that is commonly used and referred to because it is a non-negative number. It is the principal square root and typically what most calculators and educational contexts refer to when they mention 'the square root'.

In general, for any non-negative number 'a', the positive root is the non-negative number 'b' such that\[b \times b = a\]For 36, verifying the positive root involves checking:

• Evaluate whether \(6 \times 6 = 36\)
• Since 36 equals 36, it confirms that 6 is indeed the positive root

This method of evaluation proves the square of 6 brings us back to the original number of 36.

Just like a number can have a positive root, it can also have a negative root. While the term "square root" often brings the positive root to mind, every positive number actually has two roots: a positive and a negative one.

In our example with the number 36, we also find that \[-6\]is a viable square root. This is because when you multiply two negative numbers, the negatives cancel out, resulting in a positive product. Thus,\[(-6) \times (-6) = 36\]The negative root essentially mirrors the positive one but in the opposite direction on a number line.

To reinforce understanding:

• Multiply \(-6 \times -6\)
• The outcome equals 36, verifying \(-6\)
• This shows that both roots balance the equation squarely to the initial number

This duality reminds us that `square root` encompasses two solutions whenever we're dealing with positive numbers.

Verifying whether numbers are roots of a given value involves straightforward arithmetic. It's essential to be certain about both potential positive and negative roots to claim a comprehensive understanding of square roots.

To verify, you can use simple multiplication tests. As we've seen, both 6 and -6 satisfy the condition that multiplying each by itself results in 36. This thorough verification process removes any ambiguity:

• For the positive root, check that \(6 \times 6 = 36\)
• For the negative root, confirm that \(-6 \times -6 = 36\)

Both conditions must satisfy the original number, ensuring both are valid square roots.

This concept reinforces the necessity of considering multiple answers in square root problems and provides tools to methodically prove both positive and negative solutions are indeed valid.