Problem 1 - Elementary Algebra

Question

Name both square roots of each of the following numbers. $$ 36 $$

Step-by-Step Solution

Answer

Answer: The two square roots of 36 are 6 and -6.

Step 1: Find the positive square root of 36.

To find the positive square root of a number, think of a number that, when multiplied by itself, results in the given number. We know that $$6 \times 6 = 36$$, so the positive square root of 36 is 6.

Step 2: Find the negative square root of 36.

Since we are looking for both square roots, we must also find the negative square root. We know that $$-6 \times -6 = 36$$, so the negative square root of 36 is -6.

Step 3: Write the final answer.

Both square roots of the given number 36 are 6 and -6.

Key Concepts

Positive Square Root

The positive square root of a number is the non-negative number that, when multiplied by itself, equals the given value. For instance, the positive square root of 36 is 6. This is because when you multiply 6 by itself, you get 36:

• 6 × 6 = 36

Every positive number actually has two square roots: one is positive and the other negative. But the term "square root" often refers to just the positive square root. In mathematical notation, the positive square root of a number \( x \) is often represented as \( \sqrt{x} \). For 36, this would look like \( \sqrt{36} = 6 \).
The square root is a fundamental concept in math often used in algebra, geometry, and calculus. It's important to remember that we usually consider only the positive square root unless specified otherwise.

Negative Square Root

While the positive square root provides a non-negative solution, there is also a negative square root for any positive number. This negative square root is simply the negative of the positive square root. For example, the negative square root of 36 is -6:

• -6 × -6 = 36

Even though we usually refer to the positive square root by default, both positive and negative values satisfy the definition of the square root. The negative square root is useful in quadratic equations and other mathematical problems that involve symmetry. Mathematically, the negative square root of a number \( x \) can be denoted as \( -\sqrt{x} \). In the case of 36, this looks like \( -\sqrt{36} = -6 \).
Recognizing both positive and negative roots is important when solving equations because they provide a complete set of solutions.

Multiplication

Multiplication plays a vital role in understanding square roots. A square root essentially asks: "What number, when multiplied by itself, gives the original number?" Multiplication of integers, like 6 times 6 or -6 times -6, is straightforward.
In multiplication, the product of two positive numbers is positive, and the product of two negative numbers is also positive. This is why both \( 6 \times 6 \) and \( -6 \times -6 \) equal 36.

• This multiplication principle also implies that neither a positive nor negative number can result in a negative product when multiplied by itself.
• This concept ensures that the square roots of any positive number will be both positive and negative.

Multiplication is not just about basic arithmetic—it is the foundation through which we explore mathematical patterns and solve complex problems, including those involving square roots.