Problem 31 - Algebra 1

Question

Simplify the expression. $$\sqrt{3} \cdot \sqrt{12}$$

Step-by-Step Solution

Answer

The simplified expression is 6.

Step 1: Identify the manipulation technique needed

The expression presented can be simplified by applying the multiplication rule of square roots, which states that \( \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \). Here \(a = 3\) and \(b = 12\).

Step 2: Apply the multiplication rule

Multiply 3 and 12 under the square root to get \( \sqrt{3 \cdot 12} = \sqrt{36} \).

Step 3: Simplify the square root

The square root of 36 simplifies to the number 6, because 6 squared \( (6^2 = 36) \) equals 36.

Key Concepts

Simplifying Square Roots

Understanding how to simplify square roots can make expressions easier to work with and solve. Square roots often involve numbers that aren't perfect squares, like 12, which can't be simplified directly. However, we can break it into factors. For example, 12 can be factored into 4 and 3, where 4 is a perfect square. When simplifying a square root, such as \( \sqrt{12} \), splitting it into \( \sqrt{4} \times \sqrt{3} \) helps. Here, because \( \sqrt{4} = 2 \), the expression simplifies to \( 2\sqrt{3} \).

To put it simply:

• Factor numbers inside a square root if possible.
• Extract perfect squares.
• Combine them outside the square root.

Practice breaking down various square roots to see how this technique works in different scenarios.

Product Property of Square Roots

The product property of square roots is a convenient rule that simplifies complex multiplication under square roots. It states: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} \). This property is powerful because it allows you to combine separate square roots into one.

Using this property, the expression \( \sqrt{3} \cdot \sqrt{12} \) becomes \( \sqrt{3 \times 12} = \sqrt{36} \). The only key step is performing the multiplication inside the square root before simplifying.

• This property only works for non-negative numbers \(a\) and \(b\).
• Always check if the product under the square root can be further simplified.

Mastering this property helps in making operations with square roots more manageable and fewer steps are required in calculations.

Basic Algebraic Expressions

Basic algebraic expressions often involve operations like addition, subtraction, multiplication, and division, sometimes with square roots included. Understanding these expressions involves knowing how to handle variables and constants.

In the context of square roots, algebraic expressions can appear complicated, but simplification is key. For instance, in our example \( \sqrt{3} \cdot \sqrt{12} \), identifying that there are no variables makes it easier to apply simple arithmetic and square root properties.

• Look for elements you can combine or simplify.
• Practice performing operations while respecting properties of square roots and numbers.
• Verify your solutions by re-evaluating the rationale behind each step.

Developing these skills leads to stronger foundational understanding in algebra, crucial for advancing in mathematics.