Problem 27 - Elementary Algebra

Question

Simplify each expression by removing the radical sign. Assume each variable is nonnegative. How many square roots does every positive real number have?

Step-by-Step Solution

Answer

Answer: Every positive real number has two square roots, one positive and one negative.

Step 1: Understand the nature of a square root

A square root is a number that, when multiplied by itself, results in the original number. For example, the square root of 9 is 3 because 3 * 3 = 9. Removing the radical sign means finding the square root of the given expression.

Step 2: Determine the number of square roots for positive real numbers

As we need to find how many square roots does every positive real number have, consider it this way: Let's take x is a positive real number, then we need to find the square roots of x. There are two possibilities for a square root of x: one positive and one negative. For example, the square roots of 4 are +2 and -2 because (+2) * (+2) = 4 and (-2) * (-2) = 4. Now let's summarize our findings:

Step 3: Conclusion

Every positive real number has two square roots, one positive and one negative.

Key Concepts

Radical Sign

The radical sign, represented as \( \sqrt{} \), is a symbol used in mathematics to indicate the square root of a number. When you see this sign, it prompts you to find a number which, when multiplied by itself, equals the number inside the radical. This is called 'taking the square root'.
To take the square root of a number, simply look for a number that can be squared to produce the number in the radical.

• The expression \( \sqrt{16} \) is asking for a number which squares to 16. In this case, both 4 and -4, because \( 4 \times 4 = 16 \) and \( -4 \times -4 = 16 \).
• Remember, the result of a square root operation could be positive or negative, although the radical sign conventionally refers to just the positive square root.

Understanding the radical sign is key to simplifying expressions accurately, ensuring you capture both potential values.

Positive Real Numbers

When we discuss square roots, it's crucial to understand the role of positive real numbers. These numbers, which include all the numbers greater than zero, are what we often refer to when simplifying radical expressions.
Positive real numbers naturally have two potential square roots.

• One is positive: This is commonly used in most applications.
• One is negative: While significant in some mathematical theory, it’s often ignored in basic practical calculations.

Knowing every positive real number has these two roots aids in accurately simplifying and solving square root problems in math.

Simplification

Simplification in math involves breaking down complex expressions into their simplest form. When dealing with square roots, this means either removing the radical sign or simplifying the number under it.
The process typically involves:

• Identifying the number or expression under the radical.
• Finding its square root, if possible, to express it as a product of integers.
• Leaving it under the radical sign if it can't be simplified further into a whole number.

For example, simplifying \( \sqrt{36} \) would involve recognizing that 6 is its square root. As 6 times 6 equals 36, you can express it as just 6, removing the radical sign.
The goal of simplification is to make calculations easier and solutions neater, especially in more advanced mathematics where unsimplified radicals can complicate further manipulations.