Problem 3 - College Algebra with Modeling and Visualization

Question

Simplify the expression without a calculator $$ 3(4)^{1 / 2} $$

Step-by-Step Solution

Answer

The simplified expression is 6.

Step 1: Identify the Expression

The expression given is \(3(4)^{1/2}\). This means you take the square root of 4 and then multiply the result by 3.

Step 2: Calculate the Square Root

First, find the square root of 4. The square root of 4 is 2 because \(2 \times 2 = 4\).

Step 3: Multiply the Result

Next, multiply the result from Step 2 by 3. So, you have \(3 \times 2 = 6\).

Key Concepts

Square Roots

Square roots are all about finding which number, when multiplied by itself, gives you the original number. So, for example, the square root of 4 is 2, because when you multiply 2 by itself, you get 4 (i.e., \(2 \times 2 = 4\)).

Square roots are important in simplifying expressions because they help us break down numbers into more manageable units. The notation for the square root is \(\sqrt{}\), and in algebraic expressions, it’s often written as \(x^{1/2}\) or \((x)^{1/2}\).

• The square root of 9 is 3 since \(3 \times 3 = 9\).
• The square root of 16 is 4 because \(4 \times 4 = 16\).
• The square root of 25 is 5 as \(5 \times 5 = 25\).

Always remember that the square root of a perfect square (like 4, 9, 16, 25) is a whole number, making calculations easier during simplification steps.

Multiplication

Multiplication is one of the fundamental arithmetic operations that combine equal groups of numbers. When you see a multiplication sign, know it means 'groups of.' In our given problem, we encounter multiplication when we need to "times" the square root result by 3.

In our example \(3 \times 2 = 6\), this shows you have three groups of two, which is essentially adding 2 three times (2 + 2 + 2 = 6). Here’s how multiplication simplifies expressions:

• When you multiply numbers, you're combining them into a single number, which simplifies calculations.
• Multiplication is commutative, which means you can change the order of numbers: \(4 \times 3 = 3 \times 4\).
• It’s associative too, so you can group numbers in any way: \((2 \times 3) \times 4 = 2 \times (3 \times 4)\).

Multiplication is crucial for simplifying algebraic expressions because it lets you quickly scale values up or down as required.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operations. They are the building blocks of algebra and can represent real-world situations.

In our task, we simplify the expression \(3(4)^{1/2}\). Here's how algebraic expressions function in general:

• A typical algebraic expression could be \(2x + 3\), where \(x\) is a variable.
• Expressions can be simple, like a single number or variable, or more complex ones with multiple operations.
• Simplifying means reducing the expression to its most manageable form while retaining its value.

Learning to work with algebraic expressions helps us understand and solve various mathematical problems. It involves using operations like addition, subtraction, multiplication, division, and, sometimes, more advanced operations like exponents and roots to transform and simplify the expressions.