Problem 148 - College Algebra Essentials

Question

This will help you prepare for the material covered in the next section. a. Find \(\sqrt{16} \cdot \sqrt{4}\) b. Find \(\sqrt{16 \cdot 4}\) c. Based on your answers to parts (a) and (b), what can you conclude?

Step-by-Step Solution

Answer

a) The value of \( \sqrt{16} \cdot \sqrt{4} \) is 8. b) The value of \( \sqrt{16 \cdot 4} \) is also 8. Therefore, c) it can be concluded that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \) for any real numbers a and b.

Step 1: Find \(\sqrt{16} \cdot \sqrt{4}\)

First, separately find the square roots of 16 and 4. \(\sqrt{16} = 4\) and \(\sqrt{4} = 2\). Now multiply these two values. So, \(4 \cdot 2 = 8\)

Step 2: Find \(\sqrt{16\cdot 4}\)

Now, first perform the multiplication inside the square root. So, \(16 \cdot 4 = 64\). Next, find the square root of 64 which is 8.

Step 3: Comparison

Now that we have calculated both \(\sqrt{16} \cdot \sqrt{4}\) and \(\sqrt{16 \cdot 4}\), we see that they are both equal to 8. Therefore, we can conclude that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), where a and b are any real numbers.

Key Concepts

Understanding the Square Root

A square root is a mathematical concept used to find a number which, when multiplied by itself, gives the original number. For instance, the square root of 16 is 4 because 4 times 4 equals 16. Square roots are denoted using the radical symbol \(\sqrt{}\). The principal square root refers to the positive square root of a number.

Key points to remember about square roots include:

• The square root of a number \(a\) is written as \(\sqrt{a}\).
• \(\sqrt{a^2} = |a|\), meaning the square root of any square number returns the absolute value of the base.
• Only non-negative numbers have real number square roots, as negative numbers don't yield real products when squared.

Recognizing square roots is vital to understanding foundational math concepts and solving equations effectively.

Exploring the Realm of Real Numbers

Real numbers encompass all the numbers that can be found on the number line, including all the rational and irrational numbers. Rational numbers include integers, fractions, and decimals that repeat or terminate, like 3, 1.5, or \(\frac{2}{3}\).

Irrational numbers, on the other hand, cannot be written as simple fractions, and their decimal forms never repeat or terminate. Examples are \(\pi\) and \(\sqrt{2}\). In the context of square roots, real numbers ensure that expressions like \(\sqrt{16} = 4\) are meaningful.

Highlights about real numbers include:

• They can be either positive, negative, or zero.
• Operations on real numbers (addition, subtraction, multiplication, and division) follow consistent properties.
• Every real number corresponds to a point on the number line.

Understanding real numbers is essential for grasping many mathematical operations and their applications.

Multiplication of Square Roots Made Easy

When multiplying square roots, you can often simplify the expression significantly. According to the properties of square roots, \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). This property arises because both sides of the equation stem from the same mathematical principle of roots and exponents.

This principle implies that instead of multiplying two separate square roots and then simplifying, one can directly multiply the numbers inside the radical first. For example, from the given exercise, \(\sqrt{16} \cdot \sqrt{4} = \sqrt{16 \cdot 4}\), resulting in \(\sqrt{64} = 8\).

When using the multiplication of square roots, keep these points in mind:

• Ensure both numbers under the root are non-negative to keep the result within the real numbers.
• Try simplifying each root where possible before multiplication.
• This property is a time-saving tool in algebraic manipulation and equation solving.

Mastering this property helps ease complex calculations involving roots and harmonizes perfectly with other algebraic skills.