Problem 26 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{16} \cdot \sqrt{4} $$

Step-by-Step Solution

Answer

The expression \(\sqrt{16} \cdot \sqrt{4}\) simplifies to 8.

Step 1: Solve the First Square Root

First, we need to find the square root of 16. This is the number that, when multiplied by itself, results in 16. This number is 4.

Step 2: Solve the Second Square Root

Next, we will find the square root of 4. This is the number that, when multiplied by itself, results in 4. This number is 2.

Step 3: Multiply the Results Together

Lastly, we multiply the two results together. The multiplication of 4 and 2 is 8.

Key Concepts

square root

A square root asks the question: "What number, when multiplied by itself, results in the given value?" For example, finding the square root of 16 means determining which number squared equals 16. The answer is 4, because \( 4 \times 4 = 16 \). Similarly, the square root of 4 is 2, because \( 2 \times 2 = 4 \). Square roots are often denoted by the radical symbol \( \sqrt{} \), and are essential in mathematics for solving quadratic equations and simplifying expressions.
Understanding square roots can help in various math problems, especially in those involving radicals and algebraic expressions. To simplify a square root expression, identify the number that produces the original number when squared.

multiplication of radicals

When multiplying radicals, you can use the property \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b} \). This allows us to combine and manipulate radical expressions easily. For instance, when simplifying \( \sqrt{16} \cdot \sqrt{4} \), this rule can be very useful. Rather than finding each square root separately and multiplying them, we could directly compute \( \sqrt{16 \times 4} \), resulting in \( \sqrt{64} \), leading directly to 8. Multiplying radicals involves:

• Applying multiplication under a single radical if possible.
• Simplifying each step to make the calculation easier.
• Always considering if combining radicals first can provide a shortcut.

Keep in mind that this approach is valid only for non-negative numbers under the square root, as negative numbers lead to complex numbers.

basic operations with radicals

Radicals, or square roots, can be simplified and manipulated using several basic operations: addition, subtraction, multiplication, and division. Each operation has specific rules that must be followed.

• Addition and Subtraction: Can only combine radicals when they have the same radicand (the number under the radical). For example, \( \sqrt{3} + \sqrt{3} = 2\sqrt{3} \).
• Multiplication and Division: Use the properties discussed previously: \( \sqrt{a} \cdot \sqrt{b} = \sqrt{a \times b} \) and \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \) if \( b eq 0 \).

By mastering these operations, you can simplify complex expressions involving radicals. Remember to always simplify where possible to make calculations more straightforward, which will make solving math problems easier and quicker.