Problem 31 - Elementary Algebra

Question

For the following problems, find the two square roots of the given number. $$ 81 $$

Step-by-Step Solution

Answer

Answer: The two square roots of 81 are 9 and -9.

Step 1: Understanding the square root function

A square root of a number x is a number y such that y * y = x. In this case, x = 81, and we need to find two values of y, satisfying the conditions.

Step 2: Factoring the given number

Let's find the factors of the number 81. We have: - 1 * 81 = 81 - 3 * 27 = 81 - 9 * 9 = 81

Step 3: Identifying the square roots

From the list of factors above, we can see that 9 * 9 = 81. So, one of the square roots of 81 is simply 9. The second square root is the additive inverse (or opposite) of the first square root. Since the first square root is positive 9 (9), the second square root will be negative 9 (-9). Therefore, the two square roots of 81 are 9 and -9.

Key Concepts

Factorization

Factorization is an essential mathematical operation where we express a number as a product of its factors. This technique helps us break down numbers into simpler multiplicands, which is particularly useful for finding square roots.

When we factorize 81, we are looking for pairs of numbers that can be multiplied to give us 81. This approach is crucial in determining if a number is a perfect square. In the case of 81, we found through factorization that 9 * 9 = 81.

• 81 can be factorized into 1 * 81, 3 * 27, and 9 * 9.
• The pair 9 * 9 shows that 81 is a perfect square, which indicates that 9 is a potential square root.

By factorizing, we break down the process into smaller, manageable steps, thus making it easier to identify the square root. This step is crucial for both direct calculations and for verifying results.

Positive and Negative Roots

In mathematics, every positive real number has two square roots: a positive and a negative value. This happens because squaring either a positive or a negative number results in a positive product.

For 81, the positive square root is 9 because 9 * 9 equals 81.

• 9 is the positive square root of 81, denoted as \( \sqrt{81} = 9 \).
• The negative square root is -9 because (-9) * (-9) also equals 81.

This illustrates the basic principle that the square of any real number is always positive, which allows both positive and negative roots to exist. It's important to remember, whenever you're finding square roots, that both these roots are possible unless context specifies otherwise.

Mathematical Operations

Understanding basic mathematical operations is crucial when dealing with square roots. These operations include multiplication, understanding inverse actions, and exploring properties of numbers such as perfect squares.

The problem at hand involves squaring and taking square roots. Here's why it's essential:

• Squaring a number involves multiplying the number by itself (e.g., 9 * 9 = 81).
• Finding the square root is the inverse operation of squaring; it is about finding a number that, when squared, equals the original number (\( \sqrt{81} = 9 \)).

By understanding these operations, we perform these calculations with confidence and precision. This allows us to identify both the positive and negative roots efficiently. It's also a great way to verify the correctness of calculations done during factorization and root finding.