Problem 6 - Algebra A Combined Function

Question

Find the real square roots of each number. $$ 64 $$

Step-by-Step Solution

Answer

The real square roots of 64 are 8 and -8.

Step 1: Understanding Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, since \(8 \times 8 = 64\), then \(8\) is a square root of \(64\).

Step 2: Finding Positive Square Root

Identify the positive number which when multiplied by itself results in \(64\). Calculating, we find that \(8 \times 8 = 64\), so \(8\) is the positive square root of \(64\).

Step 3: Finding Negative Square Root

Identify the negative number which when multiplied by itself results in \(64\). We know that \((-8) \times (-8) = 64\), meaning \(-8\) is also a square root of \(64\).

Step 4: Listing All Real Square Roots

Combine both the positive and negative square roots. Therefore, the real square roots of \(64\) are \(8\) and \(-8\).

Key Concepts

Understanding Real Numbers

Real numbers include both rational and irrational numbers, which mean these numbers can be written as decimals. They encompass integers, fractions, and decimals that extend indefinitely without repeating. Real numbers stand in contrast to imaginary or complex numbers, providing the most familiar number type for students. Consider:

• Integers: These are whole numbers without fractional parts, such as \(-5, 0,\) or \(7\).
• Fractions: Numbers like \(\frac{1}{2}\) or \(-\frac{3}{4}\) fall into this category.
• Irrational numbers: Numbers like \(\pi\) or \(\sqrt{2}\) that cannot be expressed exactly as fractions.

Real numbers can be positive, negative, or zero. When dealing with the concept of square roots, we focus primarily on real numbers.

The Positive Square Root

The positive square root of a number is the non-negative value that, when squared, results in the original number. It is denoted with a radical symbol \(\sqrt{}\), accompanied by the number. For example, the positive square root of \(64\) is \(8\), because \(8 \times 8 = 64\). Positive square roots are fundamental in math because:

• They help in simplifying expressions.
• They appear in geometry, especially in formulas involving areas and diagonals.
• The positive square root is often the principal root used in solving equations.

It's important to remember, for any non-negative number \(a\), the square root expression \(\sqrt{a}\) refers to its positive square root. Students often default to this because in many practical applications, values can't be negative (like lengths or areas).

The Negative Square Root

Every positive number also has a negative square root, which is simply the opposite of its positive counterpart. Though less commonly used than positive square roots, negative square roots are essential for showing the complete set of solutions to equations. For instance,

• The negative square root of \(64\) is \(-8\), since \((-8) \times (-8) = 64\).
• It's usually written as \(-\sqrt{n}\) to indicate the negative square root explicitly.

Though not typically used in real-world measurements like lengths or areas (which cannot be negative), negative square roots are critical in algebra and advanced mathematics. In solving quadratic equations, recognizing both positive and negative roots ensures all potential solutions are considered. This dual nature of square roots illustrates the thoughtful balance in mathematics between opposites.